mathcounts notes

Mathcounts Chapter Prep Daily 5 Problems

2012-01-11T17:00:00.001-05:00

E-mail me your answers through comments, My students and I will check if they are right.
Have fun problem solving and good luck on your chapter competition.

#1: When x² - 3x + 2C is divided by x + 1, the remainder is -6, what is C?

#2: 13_b + 23_b = 40_b, what is base b?

#3: If you increase 5 inches to the side of the original square, the area would increase 245 square inches, what is the original area of the square?

#4: The first and the sixth number in a sequence are both 10 and starting with the third number, each number is the sum of the previous two numbers. What is the fourth number?

#5: If you spend 1/3 of the money you have on the first day, 2/5 of what you have left on the second day and 1/4 of what you have left on the third day. Now you have 12 dollars left. How much do you originally have?

2009 Mathcounts School Sprint

2012-01-09T11:41:00.001-05:00

#17. What is the sum of the positive multiples of 13 that are less than 300?

#19. What is the smallest whole number that has a remainder of 1 when divided by 4, a remainder of 1 when divided by 3, and a remainder of 2 when divided by 5?

#20. A palindrome is a number that reads the same forwards and backwards. The sum of a particular set of three consecutive positive integers is a three-digit palindrome. If the sum is less than 220, what is the greatest possible value for the largest of the three integers in the set?

#26: The isosceles trapezoid shown has side lengths as labeled. How long is segment AC?

#27: The perimeter of a particular square and the circumference of a particular circle are equal. What is the ratio of the area of the square to the area of the circle? Express your answer as a common fraction in terms of π.

28. For a set of five positive integers, none greater than 100, the mean is 1.5 times the mode. If 31, 58, 98, x and x are the five integers, what is the value of x?

29. Square ABCD has side lengths of 13 units. Point E lies in the interior of the square such that AE = 5 units and BE = 12 units. What is the distance from E to side AD? Express your answer as a mixed number.

30. Given any positive even integer, x, the positive difference between the smallest odd number greater than 7x − 2 and the largest odd number less than 3x + 5 can be written in the form ax + b. What is a + b?

Solutions:

#17: The sum of any arithmetic sequence is average times the terms (how many numbers in that sequence).

There are 23 numbers from 13 x 1 = 13 to 13 x 23 = 299 and their average is (13 + 299) / 2 = 156.

23 x 156 = 3588.

#19: 1 , + 3 x 4 , + 3 x 4 x 2, , + 3 x 4 x 3, etc... are all numbers when divided by 3 or 4 will leave a remainder of 1.

Those numbers when divided by 5 will leave a remainder that looks like this:

1, 3, 0, 2, 4, 1, 3, 0...

1 + 3 x 4 x 3 = 37

#20: The sum of a particular set of three consecutive positive integers is a three-digit palindrome ≤ 220.

Since all three are positive consecutive integers, the sum has to be evenly divide by 3.

The greatest possible value is 171. 171 / 3 = 57 so the largest number is 58.

#26: See the image for solution.The answer is 17 units.

#27: Let the side length of a particular square be s and the radius of a particular circle be r and you have

4s = 2rπ.

s = 1/2 of r π or rπ / 2.

Their area ratio is ( rπ / 2)²/ πr² = π / 4.

#28: x is the mode and according to the given, we can set up the following equation:

( 187 + 2x ) / 5 = 1.5 x, 187 = 5.5 x, x = 187 times 10/55 = 17 x 2 = 34

#29: See image and explanations for solution.

#30: x is and even integer so 7x - 2 is also even (even - even = even).

The smallest odd number greater than 7x - 2 is 7x -1.

3x + 5 is odd (even + odd = odd) so the largest odd number less than

3x + 5 is 3x + 3.

7x - 1 - (3x + 3) = 7x - 1 - 3x - 3 = 4x - 4.

a = 4 and b = -4 so a + b = 0.

2010 Mathcounts School Sprint

2012-01-05T17:06:00.001-05:00

Solutions are below; however, refrain yourself from reading them too soon.
Happy problem solving!!

#21: Ms. Osborne’s class wants to decorate its classroom, and the students want to collect the money to fund the decorating. If every student gives $1.50, then the class will be 95¢ short. If every student gives $1.60, then the class will have $1.15 left over. How many students are in Ms. Osborne’s class?

#22: A bag contains exactly three red marbles, five yellow marbles and two blue marbles. If three marbles are to be drawn from the bag at the same time, what is the probability that all three will be the same color? Express your answer as a common fraction.

#23: A target consists of concentric circles of radii 1 cm, 2 cm and 3 cm. The innermost circle is colored red, the middle ring is colored white, and the outer ring is colored blue. If a point is chosen at random on the target, what is the probability that it lies in the blue region? Express your answer as a common fraction.

#24: Fido can chew a strip of rawhide at a rate of five inches per minute. Fluffy can chew a strip of rawhide at a rate of two inches per minute. Fluffy starts chewing on a long strip of rawhide, and six minutes later, Fido starts chewing on the other end. Each of them chews until all of the rawhide is gone, and in the end they have each consumed half of the original piece of rawhide. How long was the original strip of rawhide?

#25: A student correctly answers 15 of the first 20 questions on an examination. He then answers 13 of the remaining questions correctly. All of the questions are worth the same amount. If the student’s final score is 50%, how many questions are on the exam?

#26: A square and an equilateral triangle have equal perimeters. The area of the triangle is 163 square centimeters. How long, in centimeters, is a diagonal of the square? Express your answer in simplest radical form.

#27: Six students (four juniors and two seniors) must be split into three pairs. If the pairs are chosen randomly, what is the probability that the two seniors form one pair? Express your answer as a common fraction.

#28: Bus A is 150 miles due east of Bus B. Both busses start driving due west at constant speeds at the same time. It takes Bus A 10 hours to overtake Bus B. If they had started out at the same time, had driven at the same constant speeds, but had driven toward one another, they would have met in 2 hours. What is the speed, in miles per hour, of Bus A?

#29: Let the function a @ b be defined as 3a + 2b for all real numbers a and b. If u and v are real numbers for which u @ v = v @ u = 20, what is the value of 2u @ 3v?

#30: If a positive two-digit integer is divided by the sum of its digits, the quotient is 2 with a remainder of 2. If the same two-digit integer is multiplied by the sum of its digits, the product is 112. What is the two-digit integer?

Solutions:

#21: Let there be x students in Ms. Osborn's class.
150 x + 95 (cents) = 160 x -115; 210 = 10 x; x = 21 students

#22: There are 3 Red, 5 Yellow and 2 Blue marbles.
To have all three of the same color, we can only have either 3 Red or 3 Yellow.
Without replacement the probability is 3/10 times 2/9 times 1/8 + 5/10 times 4/9 times 3/8 = 66/ (10 x 9 x 8)
Simplify and you have 11/120.

#23: (3²π - 2²π) / 3²π = 5/9

#24: Let x be half of the rawhide. Six minutes later, Fluffy would choose 2 x 6 = 12 inches of rawhide and its other half now only has (x - 12) inches left.

(x - 12) / 2 = x/5, x = 20 so the original strip of rawhide is 2x or 40 inches long.

#25: Let x be the remaining questions.
15 + 1/3 of x / (20 + x) = 1/2 ; Cross multiply and you have 30 + 2/3 of x = 20 + x
Both sides times 3 to get rid of the fraction: 90 + 2x = 60 + 3x; x = 30
20 (first 20 questions) + 30 (remaining questions) = 50 total questions

#26: Let S be the side of the equilateral triangle.
√ 3 /4 times S² = 16√ 3 ; S = 8; Perimeter of the triangle = 8 x 3 = 24 and one side of the square
= 24 / 4 = 6 so the diagonal of the square = 6√2 (45-45-90 degree angle ratio)

#27: Solution I: There are (6C2)(4C2)(2C2) / 3! (order doesn't matter,thus divided by 3! since there are 3 groups of pairs.) = 15 groups total. If you have the two seniors in one group, then there will be 4C2(2C2)/ 2! = 3 ways to pair them. 3/15 = 1/5

Solution II: If you pick one of the two seniors, there are 5 students left and the probability that the two seniors form one pair is 1/5.

#28: Let A be the rate of Bus A and B be the rate of Bus B. You can set up the following equations according to the given information.
(A-B) times 10 = 150 so A - B = 15
(A+B) times 2 = 150 so A + B = 75
Add the two equation together and divided by 2 and you have A = (15 + 75) / 2 = 45 mph

#29: According to the rule, u@v = 3u + 2v = 20 and v@u = 3v + 2u = 20 as well. Plug the two equations
2u@3v = 6u + 6v = 6 times 8 = 48

#30: Solution I:
Let the two digit numbers be 10a + b and you have
(10a+b) = 2(a + b) + 2 so 8a - b = 2, b = 8a - 2
(10a + b) (a + b) = 112
Plug in b = 8a -2 and you'll get (18a -2) (9a-2) = 112
18a times 9a - 54a + 4 -112 = 0; Dividing all the equation by 54 and you have 3a² - a -2 = 0
(3a + 2)(a -1) = 0 so a = -2/3 (discard) or 1; b = 6 so the two digit number is 16.

Solution II;
(10a+b) = 2(a + b) + 2 so 8a - b = 2, b = 8a - 2a
(10a + b) (a + b) = 112
a and b are both ≤ 9, Factor and equation and you have (10a + b)(a + b) = 112 = 16 x 7
a = 1 and b = 6 (It works!!) The answer is 16.

Mathcounts competition preparation strategies I

2011-12-28T13:45:00.000-05:00

It's time to think about the upcoming Mathcoutns chapter and state competitions.

Here are some links/sites to get you started.

Mathcounts Mini : At the very least, finish watching and understanding most of the questions from 2010 through 2012, since detailed solutions are provided along with some more challenging problems.

For those who are aiming for the state competition, you can skip the warm-up and go directly to "The Problems" used on the video as well as work on the harder problems afterward.

For concepts reviewing, try the following three links.

Mathcounts Bible

Mathcounts Toolbox

Coach Monks's Mathcounts Playbook

You really need to understand how each concept works for the review sheets to be useful.

To my exasperation, I have kids who mix up the formulas without gaining a true understanding and appreciation of how an elegant, seemingly simple formula can answer myriads of questions.

You don't need a lot of formulas, handbook questions, or test questions to excel.

You simply need to know how the concepts work and apply that knowledge to different problems/situations.

Hope this is helpful!!

Happy holiday, Mrs. Lin

2011 Mathcounts Chapter Sprint Round solutions

2011-12-14T09:57:00.000-05:00

#22: The answer is 2674.
See left for explanations.

#23: Let the two consecutive positive integers be x and x + 1.
( x + 1 ) / x = 1.02, x + 1 = 1.02x, 0.02x = 1, x = 1 divided by 0.02 = 1 times 100/2 = 50
The two numbers are 50 and 51 and their sum is 50 + 51 = 101.

#24: The two x-intercepts when y is "0" are 10 or -10; the two y-intercepts when x is "0" are 5 or -5.
Area of a rhombus is D1 x D2 / 2 so the answer is [10-(-10)] x [5 -(-5)] = 100 square units.

#25: The area ratio of the two similar triangle is 150/6 so the line ratio is √ 150/6 or 5:1.
the length of the hypotenuse of the smaller triangle is 5 inches, so the other two legs are 3 and 4. (a Pythagorean triple)
The sum of the lengths of the legs of the larger triangle is (3 + 4) x 5 = 35.

#26: To have same number of boys and girls, the committee needs to consist of 3 boys and 3 girls.
(6C3 x 4C3)/ 10C6 = 80/210 = 8/21

#27: When the point (3, 4) is reflected over the x-axis to B, B would = (3, -4).
When B is reflected over the line y = x to C, C would = (-4, 3).

The area of the triangle is [4 -(-4)] x [ 3 - (-4)]/ 2 = 28 square units.

#28: Tonisha is 45 miles ahead Sheila when Sheila leaves Maryville at 8: 15 a.m. Each hour Sheila
will be 15 miles closer to Tonisha. 45/15 = 3, which means that 3 hours later Sheila will pass Tonisha.
8:15 + 3 hours = 11: 15 a.m.

#29: Using 30-60-90 degree angle ratio, you can
make the radius be √ 3 and half of the side of the hexagon would be 1 so each side of the hexagon
is 2.

The area of the hexagon is (√ 3/4) times 2² times 6 = 6√ 3.
The area of the circle is 3Π.
The fraction is 3Π/6√ 3 = √ 3 / 6
a = 3 and b = 6, ab = 18

#30: Area of triangle KDC is easy to find once you realize the height is just the right triangle with a hypotenuse 6 and a leg 4. (half of the length of CD),

Using Pythagorean theorem, you get the height to be 2√ 5 .

8 x 2√ 5 /2 = 8 √ 5 .

Notes on 2011 Mathcounts School Team Round Questions

2011-12-10T20:58:00.003-05:00

#1: 20 + 75 times 0.05 = 23.75

#2: The original line is 2x - 3y = -3. When you reflect a point across the x-axis, the x-coordinate remains the same, but the y-coordinate is transformed into its opposite.
2x - 3y = -3 , multiply by -1 and you have 3y -2x = 3, y = 2x +3/ 3 = 2/3 times x + 1
The new line is y = -2/3 of x -1, m = -2/3 and b = -1
m + b = (-2/3) -1 = -5/3

Here is the link for further study: Reflections: Lines and Points.

#3: If each side of the square is 20 inches long, the radius of the circle is 20/4 = 5
20² - 4 x 5²π ≈ 85.84 sq inches

#4: Let x be Mickey's bike-riding speed. x times 30/60 (convert the time to hour) and rate times
time = distance = 6 miles, x = 12 mi / hour

Let y be Mickey's walking speed. 5/12 (distance divided by speed = time) + 1/y = 45/60 (speed is in hour so you need to make the unit consistent.)
Times 12 y on both sides and you have 5y +12 = 9y, 4y = 12, y = 3 mi / hr

#5: (4.5 + 1.25 + 1.85) + ( 4.5 + 0.75 + 1.85) + ( 4.5 + 1.25 + 1.1) - 1.5 = 20.05 dollars

#6: (6 + x + 22)/ 3 = x, Times 3 on both sides: 28 + x = 3x, x = 14

#7: Since the area of the square is 81 square units, the side length is 9. Dividing each side into 3 equal parts,
each smaller part is 9/3 = 3. There are 3 x 4 = 12 for the middle sections.
The four-side arch length, if combined, is a full circle with a radius 3, or a diameter 6.
12 + 6π ≈ 30.8 units

#8: Since 4 and 7 are relatively prime, the easiest way to solve this problem is to do 4 x 7 - 4 - 7 = 17.
Or you can list the possible scores: 4, 7, 8, 11, 12, 14, 15, 16, 18, 19, 20, 21 (stop)
Once you have 4 consecutive numbers, all the other scores can be obtained.
The answer is 17.

#9: Let there be x coins. According to the given, the equation looks like this:
[20x (average times the terms = sum) + 25] / (x + 1) = 21
20x + 25 = 21x + 21, x = 4, If there are 4 coins and the total is 20 x 4 = 80, there are 3 quarters. (and 1 nickel)

#10: (2√ 2 /3 + √ 2 /3) x 2(√ 2 )/4 =
√ 2 x 1/2√ 2 = 2 / 2 = 1 sq units
You don't need to divided by two because there are two congruent trapezoids.

Notes on 2011 Mathcounts School Target Round Questions

2011-12-09T09:35:00.002-05:00

#1: To spend the least amount of money on cupcakes, you want to buy the bundle packages.
16 x 8 = 128, so you can buy 10 packages of 12, which would cost 10 x 16 = 160
You need to buy another package of 6 cupcakes + 2 extra cupcakes, which costs total 8.5 + 3 = 11.5
The answer is 160 + 11.50 = 171.50 dollars.

#2: Only when 2 is on the table hidden, the sum will be odd, so 1/6.

#3: If the eighth graders sold 2x gift bags, according to the given information, they would sell x rolls of wrapping paper.
2x times 1.5 + 2.5 times x = 704,     5.5x = 704,    x = 704 divided by 5.5 = 704 x (10/55) = 704 x (2/11)
= 128 --704 divided by 11 = 64
2x = 2 x 128 = 256 gift bags

#4: to figure out the percentage increase/decrease, you find the positive difference/ the original amount
[(1.2)/5 - (1.2)/6 ] over (1.2/6) = .2 = 20%

#5: First of all, you need to learn triangle inequality.
To do this question fast, break the three legs into sum of two and another leg.
Since according to the triangle inequality, the sum of any two legs has to be larger than the third leg.
You can have the following possible combinations.
8 - 7: In this case, the triangles can be 1-7-7, 2-6-7, 3-5-7, and 4-4-7.
9 - 6: In this case, the triangles can be 3-6-6, 4-5-6
10 - 5: In this case, the triangles can be 5-5-5
There are 7 triangles.

#6: 19B + 5O    = 35 --- equation 1
        6B + 10O = 35 ---equation 2
Add up the two equations and you have 25B + 25O = 70 so it's 70-35 = 35 dollars more than 35 dollars.

#7: The relationship between hourly work and earning is direct.
Jackie: Karen: Lex = 5 and 1/2 : 3: 1 and 1/2 = 11 : 6 : 3 (times 2 for the previous ratio)
Jackie should earn 11/20 (parts over total )times 75 = 41.25 dollars.

#8: The answer is 144 sq. feet.

Another way is to use algebra:

(x +6)(x + 16) - x(x + 10) = 192
x² + 22x + 96 -x² -10x = 192
12x = 96, x = 8
8 x 28 = 144 sq. feet

Notes on 2011 Mathcounts School Spring Round Questions

2011-12-08T21:15:00.001-05:00

#6: You should know how to do fraction of a number = another number fast.
In this case, 3/4 of x = 4.5, x = 4.5 x (4/3) = 6 feet

Other question such as 3/5 of a number = 15 what is 2/5 of a number. You don't need to find the number.
Since the denominator is the same, you can delete it and think of this question as 3 parts is 15, what is 2 parts.
The answer is 10.

#9: Us as fewer variables as possible when solving algebra questions.
Let there be x numbers of student tickets, then there are (45 - x) numbers of adult tickets.
5x + 7 (45 -x) = 265, 5x + 315 - 7x = 265, -2x = -50, x = 25 student tickets

11. There are 60 minutes in an hour and 360 degree in a circle, so one minute = 360/60 = 6 degrees.
810 / 6 = 135 minutes.

You can also use 810 /360 = 9/4 hours. 9/4 times 60 = 135 minutes.

#14: After simplifying you have 5! = 5 x 4 x 3 x 2 x 1 and 8!/3! = 8 x 7 x 6 x 5 x 4
Both have 5 x 4 x 3 x 2 so the answer is 120

#15: When it's product = a number, a lot of the times it's factoring question.
3705 = 741 x 5 = 247 x 3 x 5, One of the teens is "15".
247 the unit digit is 7 so the two teens are 13 and 19. 13 x 19 = (13 + 9) x 10 + 3 x 9 = 247 so
the three teens are 13, 15, and 19 years old. Their sum is 47.

#16: 100 6th grade girl buy lunch and 70 6th grade boys bring lunch (given).
There are 160 boys in the 6th grade, which means (160 - 70) = 90 boys buy lunch.
There are 160 6th grades bring lunch so 160 + 100 (6th grade girls who buy lunch) + 90 (6th grade boys who buy lunch) = 350 students

#17: Largest 3-digit perfect square is 31² = 961 and largest three-digit
perfect cube is 9³ = 729 so their average is (961 + 729) / 2 = 845

#18: Let Billy Goats invested x dollars in bonds, then he invested 4.5x dollars
in stocks. 4.5x / 5.5x times 165000 = 9/11 times 165000 = 9 x 15000 = 135,000 dollars.

#19: 45% times 40/(40 + 10) = 18/50 = 36%

#20: The midpoint of (4, 0) and (4, 8) is (4, 4,), which is the center of the
circle O. The midpoint of (13, 0) and (13, 10) is (13, 5).
The slope of two points (4, 4) and (13, 5) = (5-4)/(13-4) = 1/9

#21: There are total 5x5x5 = 125 unit cubes. There are 8 x 6 = 48 unit cubes that will have one face painted. There are 4 x 2 = 8 unit cubes that will have 3 faces
painted. 125 - (48 + 8) = 125 - 56 = 69 unit cubes that have no paint on them.
The fraction is 69/125

#22: 1-2 or 2-1 two ways; 2-3 or 3-2 two ways; 3-4 or 4-3 two ways, 4-5 or 5-4 two
ways and last 5-6 or 6-5 two ways. The answer is 2 x 5 = 10, 10/36 or 5/18

#23: 21/99 = 7/33= a /b so a = 7 and b = 33, b - a = 33 - 7 = 26

#24: 6 x 6 (area of the square) - 3²π = 36 - 9π, a = 36 and b = 9, 36 + 9 = 45

#25: x + y = 11 and x² - y² = (x + y)(x - y) = 55
so x - y = 5

#26: A 4 x 4 x 4 unit cubes will have 2 x 2 x 2 = 8 cubes that are hidden inside.
Since it's a hollow cubes, you do 64 - 8 = 56 unit cubes.

#27: 9 : S = S : T = 3, S = 3 and T = 1, The shaded are is 1 x 1 x 8 + 3 x 3
= 17 sq. feet

#28: 14_b + 24_b = 41_b, 1b + 4 + 2b + 4 = 4b + 1
3b + 8 = 4b + 1, b = 7, so base 7.

#29: The top and bottom surface = 2 times 6²π - 2²π = 64π
The outer surface = 6 x 2 x π x 10 = 120 π, The inner surface area = 2 x 2 π x 10 = 40π
Add them up and you have 64π + 120π + 40π = 224π

#30: x² - 5x + 3C = (x -3) times Quotient -12
Substitute x = 3 and you have 3² - 5 x 3 + 3C = -12, 3C = -6, C = -2

2011-2012 Mathcounts Similarity Stretch

2011-12-07T11:26:00.001-05:00

#293: DE // BC , ∆ADE ∼ ∆ABC. DE/BC = AD/AB, AB= 12 so BD = 9.

#294: The equivalent side of the larger pentagon is twice the size of smaller pentagon.
The perimeter of the smaller pentagon is ½ (7 + 8 + 3 + 5 + 3) = 13

#295: Triangle ARS, AVW and AUT are similar
because 3/12 = 12/48. and RS // UW // UT.

Trapezoid RSWV ∼ VWTU.

XW = 10, YT.= 4 x 10 = 40

#296: ∆PKN ∼ ∆PLM because AAA. PN / KN = PM / LM
PN / 8 = 6/10, PN = 4.8

#297: ∆YUV ∼ ∆YXZ, YV =3 and YZ = 5 (given) so the area ratio of the two triangle is 3² : 5² = 9 : 25
The area ratio of ∆UYV to trapezoid is 9 : (25-9) = 9 /16

#298: The answer is 144 unit²

#299:The answer is 7/19

#300: The volume ratio of the two 3-D structure toy houses is 1000 to 216, so their area ratio is
1000^2/3 to 216^2/3 or 100 to 36.
The surface area of the larger house is 400 so the surface area of the smaller house
is 36 x 4 = 144 cm²

#301:The answer is 4.

#302: The answer is ab/ (a + b)

2011-2012 Mathcounts Right Triangle Stretch

2011-12-06T21:15:00.004-05:00

#271: Use Pythagorean Theorem: a² + b² = c²
Also, learn those common triples by heart.

3-4-5 and its derivative: 6-8-10. 9-12-15. 12-15-20, 15-20-25

5-12-13, 10-24-26

8-15-17

7-24-25

9-40-41
12-35-37
20-21-29
4² + 9² = c², c = √ 97 = 9.8

#272: It's one of the triples: 9-12-15

#273: x = √ 39 = 6.2

#274: In an isosceles triangle, the height will break the base into two equal length and half of the base
is 5 so this is another triple : 5-12-13. x = 12

#275: x is the space diagonal, which can be solved by square root the sum of the square of each of the three
dimensions.
x = √ 18² + 24² + 30² = √1800 = 42.4

#276: Another triple: 9-40-41

#277: 45-45-90 degree angle ratio is 1: 1: √ 2
so the answer is 5√ 2 = 7.1

#278:30-60-90 degree angle ratio is 1: √3: 2
60 degrees is 4√3 so 30 degree is 4 and 90 degree is 4 x 2 = 8

#279: Each side is 8. If you draw a line connecting one vertice to the center x is the side opposite 60 degree.
The length opposite 30 degree is 1/2 of 8 or 4 so the answer is 4√3 = 6.9

#280: 45-45-90 degree angle ratio.
The Length opposite 90 degree is 6 so x is 6/√ 2 = 3√ 2 = 4.2

#281: Pythagorean triple 30-34-16 (derivative of 15-17-8) so y = 16
x is 3 times 34 = 102.

#282: Triangle ACD is similar to triangle ABC. AC/AD = AB/AC = 10/6 = 6/x
x = 3.6. Using the same method y = 4.8

2011-2012 Mathcounts workout 9

2011-12-06T10:05:00.003-05:00

#261: The answer is 241.3

#262: Let a students wear blue jerseys and b students wear red jerseys.
There are two cases to consider: one is 2 plays wear blue and 1 wears red or 2 wear red and 1 wear blue.
bC2 x aC1 + bC1 x aC2 = [b x (b - 1)]/(2!) times a + b times [a(a-1)]/2! = 25
Both sides times 2 and you have ab² - ab + ba² - ab = 50 , ab (b-1) + ab (a -1) = 50
ab (b - 1 + a -1) = 50, ab(a + b -2) = 50 , Factor "50" and you see ab = 10 and a + b -2 = 5
works if the two numbers are 2 and 5. The answer is 2 + 5 = 7

#263: If 13 years ago the price is y, after 13 years, with the ticket price increases 4% each year, it would
be y x ( 1.04)¹³.
y/y x ( 1.04)¹³ = 0.60057 The answer is 60%

#264: Let the three sides be x, xr, and xr² and you want to find what
is x/xr², or 1/r²
Using Pythagorean theory, you have x² + (xr )² = (xr²)²
x² + x²r² = x²r⁴ Divide both
sides by x² and the equation becomes 1 + r² = r⁴
r⁴ - r² -1 = 0, r² = 1 + √ 5 / 2
1/r² = 1/ (1 + √ 5 / 2) = 0.62111 The answer is 0.62

#265:This one is very straightforward. The area of the rectangle is (3 + 3) x 3 =18
Minus the 2 quarter circle: 18 - 1/2 of (3² Π) = 3.8628... The answer is 3.9

#266: A number ends in 2 or 7 if when divided by 5, leaves a remainder of 2.
So check, 102, 107, 112, 117. etc.. to see if it leaves a remainder of 1 when divided by 3.
The divisible rule for 3 is the sum of all the digits is divisible by 3, so in this case you want to see if
the sum will leave a remainder of 1 when divisible by 3.
The number 112 fits but it doesn't leave a remainder of 3 when divided by 7.
Use 112 and add ( 3 x 5 ) each time to see if the next number fits the criteria.
112 + 15 = 127 (no)
127 + 15 = 142 (no)
142 + 15 = 157 -- yes so the answer is 157

#267: See the solution. The answer is 2.5 units.

#268: 21 x 5 = 105 (total sum) Since the numbers are all different positive integers, the smallest median
is 3. The five numbers look like this 1 , 2, 3, ___, ___
The largest median is 1, 2, 33, 34, 35 [Use (105 - 1 - 2) divided by 3 to get the median of the larger 3 numbers. If not evenly divided, then you need to do some trial and error tests for that.].
From 3 to 33 inclusive there are 33-3 + 1 = 31 numbers.

#269: Since the time stays constant no matter what length unit you choose, we can choose the candle is 1 unit long. Let t represent the time, in hours, until the slower burning candle is twice the height of the faster burning candle.The faster candle burns completely in 6 hours so in 1 hour, 1/6 of the candle will burn. In t
hours, t times 1/6 or t/6 candle will burn leaving the length 1 - (t/6) long.
Similarly, the slower candle burns completely in 9 hours so in 1 hour, 1/9 of the candle will burn. In t hours, 1/9 of t or t/9 will burn, leaving only 1- (t/9) of the candle.
Now you can set up the equation and solve for t:
1 - (t/9) = 2 [ 1- (t/6) ] . Multiply both sides by 9 and you have 9 - t = 18 - 3t, 2t = 9 so t = 4.5 hours.

#270: Let x be the number of books Ms. Meany originally has, x + n would the total books she has later on.
(x + n) C 2 - xC2 = 99
(x + n) (x + n - 1)/ 2! - x (x -1)/2! = 99
Multiply both sides by 2 and you have (x + n) (x + n-1) - x (x -1) = 198
Consolidate and you have (x + n)² - (x + n) - x² -x = 198
Further consolidate and take out the factor and you have n (n + 2x -1) = 198 = 1 x 198 = 2 x 99 = 3 x 66
= 6 x 33 = 9 x 22 = 11 x 18
Add up all the n (those numbers in red) and you have 1 + 2 + 3+ 6 + 9 + 11 = 32

2011-2012 Mathcounts warm-up 18

2011-12-02T10:31:00.002-05:00

#251: To make the largest product using each of the digits 1,3, 5, 7, and 8 once, you will use a two-digit by three-digit numbers.
8 __ __ and 7 __ or 7 __ __ and 8 __.
The next two numbers to fit in are 3 and 5 but where. 851 x 73 = 62123
831 x 75 = 62325
751 x 83 = 62333
Using the distributive rules and you'll see why.
851 x 73 = (800 + 50 + 1) ( 70 + 3) = 800 x 70 + 3 x 800 + 50 x 70 + 3 x 50 + 1 x70 + 1 x 3
831 x 75 = (800 + 30 + 1)(70 + 5) = 800 x 70 + 5 x 800 + 30 x 70 + 30 x 5 + 1 x 70 + 1 x 5
751 x 83 = (700 + 50 + 1) ( 80 + 3) = 700 x 80 (same as 800 x 70) + 700 x 3(same as 30 x 70) + 50 x 80(same as 5 x 800) + 1 x 80 + 1 x 3
The answer is 62333

#252: Keep in mind that square is a kind of special rectangle. In any rectangle, if the length and width are closer or the same as each other, it will give you the smallest perimeter.
In this case, 20 by 20 will give you the smallest perimeter. 1/4 of that is 100 (area) so again 10 by 10 will give you the smallest perimeter. 10 x 4 = 40

#253: 13 x 7 = 91, which is the sum of the seven different positive integers.
For the median to be the smallest, you arrange them as followed:
1, 2, 3, 4, 5, 6 70, so the median is 4.
For the median to be the largest, you arrange them as followed:
1, 2, 3, ___ , ___, ___, ___
91 - 6 = 85, 85 divided by 4 is around 21. 20 is not possible because the sum of 20 + 21+ 22+ 23= 86
exceeds 85.
19 works and the rest could be 21, 22, 23 or other arrangements.
19 - 4 = 15

#254: 363 = 1 x b² + 2b + 3, b² + 2b - 360 = 0
( b-18 )(b + 20) = 0, b = 18 or - 20 The answer is 18.

#255: Square is a kind of rhombus so if you make this shape a square, you can just use 3 as each side and
BF = CG = HD = AE = 1 and EB = FC = DG = AH = 2. The side length of the smaller rhombus is
√ 5 . If the side ratio of the smaller to larger rhombus is 3 to √ 5 , the area ratio would be 3² to
√ 5 ²or 9 to 5. Area of rhombus = 5 x 180/9 = 100 unit²

#256: Plug in x = u to the first equation : ru² + su + t = 0 -- equation 1
Plug in x = 1/u to the second equation: (2 + a)times (1/u)² + 5(1/u) + (2 - a) = 0, Times u² and you have
(2 + a) + 5u + (2 - a)u² = 0, Compare this with equation 1:
t + su + ru² = 0
t = 2 + a, s = 5 and r = 2 - a. t + s + r = 9

#257: 22C4 over 22C5 = (22 x 21 x 20 x 19) / 4 x 3 x 2 x 1 over (22 x 21 x 20 x 19 x 18) / 5 x 4 x 3 x 2 x 1 = 18/5

#258: The slope of A and W is [4 - (-3)] / -9 - 2 = 7 / -11
D(-6, 2) and Z would have the same slope, so x point in Z = -6 - ( -11) = 5 and y point in Z = 2 - 7 = -5
Z = (5, -5) , 5 + (-5) = 0

#258: If you memorize enough Pythagorean triples, you'll see this one is 12-35-37 so the answer is 210 unit².
The other common triples are: 3-4-5 and it's derivatives such as 6 - 8- 10,
9-12-15 and 12 - 16 - 20.
5 - 12 - 13 , 10 - 24 - 26
7 - 24 - 25, 14 - 48 - 50
8 - 15 - 17, 16 - 30 - 34
9 - 40 - 41
12 - 35 - 37
20 - 21 - 29

Using algebra and Pythagorean theory, you will have a + c = 49 (given)
c = 49 - a
If the other leg is b, then b² = (49 -a)² - a² = 49² - 49 x 2a = 49 (49 - 2a)
49 - 2a is an odd number (why?) and a square number, Try 1, 9, 25
7-24-25 when 49 - 2a = 1 (doesn't fit)
20-21-29 when 49 - 2a = 9
12-35-37 when 49 - 2a = 25
The latter two work and both give an area of 210 unit².

#260: It's too difficult for me to draw the 3-D figure to explain this question.
E-mail me if you need solution.

2011-2012 Mathcounts warm-up 17

2011-12-02T08:08:00.000-05:00

#241: To make 15 from the set {1, 2, 3, 4, 5, 6, 7}, you can have (1, 2, 5, 7), (1, 3, 4, 7), (1, 3, 5, 6), (2, 3, 4, 6). So 1 and 3 can replace A. 1 + 3 = 4

#242: This one is pretty straightforward. The two shaded triangles together = 4 x 4 = 16.
The shaded region at the lower right hand corner = 4 x 4 - 2 x 2 = 12
The area of the upper left hand corner = 4 x 4 - 4∏ = 16 - 4∏.
Add them up and you have the answer as 44 - 4∏.

#243: Let the 5 integers be (x -2d), (x - d), x, (x + d), (x + 2d). From given x = 18 - Any arithmetic sequence the median is the same as mean.
[(x -d)² + (x -2d)² + x² + (x+d)² + (x +2d)² = x² + d² + x² + 4d² + x² + x² + d² + 4d² = 5x² +
10d²] divided by 5 = 374
(5 x 18² + 10d² ) divided by 5 = 374; d = 5
(x + 2d) = 18 + 2 x 5 = 28

#244: This one is tricky!! At the actual competition, do it later.
13
11 + 2
7 + 2 + 2 + 2
5 + 2 + 2 + 2 + 2
3 + 2 + 2 + 2 + 2 + 2
3 + 3 + 3 + 2 + 2
3 + 3 + 5 + 2
3 + 3 + 7
3 + 5 + 5
Total 9 ways.

#245: This one you need to work backward.
Xavier            Yvonne        Zeena
   48                 48            48
   24                 24                96
   12                 84                48
   78                 42                24
78 - 48 = 30

#246: Let Barbara's cell phone bill be 7x and Tina's 5x (given).
7x = 14 + 5x, x = 7
Barbara's cell phone bill = 7 x 7 = 49 dollars.

#247: 1 + 2 + 3 + 4 = 10 (not enough). We need 3 more. You can do :
1 + 2 + 3 + 7
1 + 2 + 4 + 6
1 + 3 + 4 + 5
That is it. Each has 4! or 4 x 3 x 2 x 1 = 24 ways to arrange it.
24 x 3 = 72

#248: Let the three numbers be a, b, c and you have a + b + c = 5, a² + b² + c² = 29, and abc = -10.
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
25 = 29 + 2(ab + bc + ca),   ab + bc + ca = -2
abc = -10 and ab + bc + ac = -2 (given)
abc/ ab + bc + ac = -10 / -2 = 5
c + a + b = 5, a + c = 5 - b
abc = -10 so b = -10/ ab
ab + bc + ca = ac + b (a + c) = -10/b   + b ( 5 -b) = -2 . Times b and you have -10 + 5b²-b ³ + 2b = 0
b³ -5b² - 2b + 10 = b²(b -5)-2(b-5) = 0
(b-5)(b² -2) = 0 so b = 5 , - √ 2 or √ 2
The answer is -√ 2 .

#249: According to the given information, you can set up the equation like this:
1075/1000 (25Q + 10D) = 11/10 times 25Q + 10D
Simplify and you have 43/40 (25Q + 10D) = 11/10 times 25Q + 10D
Times 40 on both sides: 43 (25Q + 10D) = 44Q times 25 + 40 x10D
43x25Q + 43x10D = 44x25Q + 40x10D
30D = 25Q, Q/D = 6/5

#250: The answer is 1008 ∏.

2011-2012 Mathcounts warm-up 12

2011-11-29T21:54:00.000-05:00

#161: Let width be x and the length is 1.5x (given). 1.5x² = 216
x² = 216 divided by 1.5 = 216 times 2/3 = 144. x = 12 -width and the length is 12 x 1.5 = 18
2 x (12 + 18 + 5 x 4) = 100 feet

#162: The answer is 3: 45 pm.

#163: Tom digs twice s fast as Dick, which means that he digs 2 holes and Dick
digs 1 hole in 6 hours. (12 holes/2) x 6 = 36 hours.

#164: From 1 to 299, there are 20 x 3 = 60 times the number "3" appears.
03, 13, 23,...93, 10 times for the unit digit and 30, 31, 32...39, another 10 times
for the tenth digit for ever hundred numbers.

From 300 to 363, there are 63 - 0 + 1 = 64 times the number "3" appears for the
hundredth digit.

303, 313,...363, or 6 - 0 + 1 = 7 times the number "3" appears for the unit digit and 330, 331, 332...339, or 9 - 0 + 1 = 10 times the number "3" appears on the tenth digit

60 + 64 + 7 + 10 = 141 times

#165:The answer is 20 squares.

#166: Median is the middle number. 36 x 1.5 = 36 x 1 + 36 x 0.5 = 54, 54 x 1.5 = 54 + 27 = 81
36 divided by 1.5 = 36 x 2/3 = 24, 24 divided by 1.5 = 24 x 2/3 = 16
(16 + 24 + 36 + 54 + 81) / 5 = 42.2

#167: According to the given information, M + 3H = 47 --first equation
In 2 years, (M + 2) = 2 ( H + 2) = 2H + 4, M = 2H + 2--second equation
Substitute M with 2H + 2 to the first equation and you have 2H + 2 + 3H = 47
5H = 45, H = 9

#168: 3O + 2C = 78---equation 1
2O + 3C = 82---equation 2
equation 2 - equation 1 and you have C - O = 4, which is the answer.

#169: If it's just one digit number, you have "5". 1 way
Two-digit numbers, you have "14, 41, 23, 32". 4 ways
Three-digit numbers, you have 113, and 122, both have three arrangements. (113, 131, 311, etc... or 3!/2!
--the denominator is how many numbers repeating.) so altogether 6 ways.
Four-digit numbers, you have 1112-- 4!/3! or 4 arrangements.
Five-digit numbers, you have 11111. 1 way
1 + 4 + 6 + 4 + 1 = 16 ways.

#170: The answer is square root of 73.

2011-2012 Mathcounts workout 8

2011-11-28T08:48:00.000-05:00

#231: 0.4% = 0.004
100 x 1.004 = 10100.4
(100.4 + 100) x 1.004=201.20
(201.20 + 100 ) x 1.004 = 302.40
... The answer is $1231.66 (I don't like this question.)

Examples of Applications of Exponential Functions

#232: Using distance formula you'll get the length of the unit to be √ (-4-4)² + (-5-2)²
= √ 113 , which is between 10 and 11. The sum is 21.

#233: 8/81 = 0.098765432
2012/9 = 223 R5, The answer is 6.

#234: 19.5 x (1.2 + 0.06)= 24.57
100%(the bill) + 20% tip = 1205 = 1.2
6% tax = 0.06

#235: There are total 3 x 10 (from 1 to 10) + 2 x 7 (from 11 to 17)+ 1 x 8
(from 18 to 25) = 52 cards.
Among all, 30 cards are 10 0r less than 10.
(30/52)(29/51)(28/50)(27/49)(26/48) about 0.055

#236: 5 √ x - 30 = 2√ x + 54, 3√ x = 84 Divided both sides by 3 and you have √ x = 28
Square both sides and you have x = 28 x 28 = 784

#237: The dimension is 1: 2 : 3 ratio and the shortest leg is 2 ft, which means the other two legs are
4 and 6. The longest distance between and two vertices are the space diagonal.
√ 2² + 4² + 6² = √56 = 7.48 ft

# 238:The answer is 30.25 cm²

#239:SAT has similar question.
(x² + ax + 6) /(x +1) = x + b (given)
x² + ax + 6 = (x + b) (x + 1) = x² + x + bx + b = x² + x (b+1) + b
b = 6 and since b +1 = a, a = 7. 6a - 7b = 0

#240:

Using 45-45-90 degree angle ratio, you get two legs of the right triangle DBC each as 3 √ 2.
CE + CB = 3√ 2 and their ratio is 2 to 1.
CE = 2√ 2 and BE =√ 2.

Using Pythagorean theorem, you have
6² = √ 2² + (3 √ 2 + x)²
34 = (3 √ 2 + x)²
√ 34= 3 √ 2 + x
x = 1.58831... about 1.59

2011-2012 Mathcounts warm-up 16

2011-11-27T07:38:00.121-05:00

#221: Let the two-digit number be 10x + y, the reversed two-digit number is thus 10y + x.
100% + 75% = 175% = 7/4 so 7/4 (10x + y) = 10 y + x
Multiply both sides by 4 and you have 70x + 7y = 40y + 4x, 66x = 33y so 2x = y.
There are 12, 24, 36, and 48, 4 two-digit numbers that fit.

#222: The play parallel to the base of the pyramid and also bisects it altitude so the height of the smaller
square pyramid is 1/2 of the larger square pyramid, and their volume is 1³
to 2³ or 1 to 8 ratio.
The volume of the smaller pyramid is 1/8 of 64 so 8 cm³.

#223: x + 4/x = y + 4/y (given) Times both sides by xy and you have
x² y + 4y = xy² + 4x
x² y - xy² = 4x - 4y, xy ( x - y ) = 4 ( x - y ), Divide both sides by ( x - y ) and
you have xy = 4.

#224: This one is more of a geometry question and you can get the area of the shaded region
pretty easily by just visually working on it.
The area of the shaded region is 6/16 or 3/8 = 37.5%
Use the smallest white unit square as a reference. The largest square is 4 x 4 = 16 and each of the
shaded region is 3/4 of a 2 by 2 square.
Also, 1/8 is 12.5% ( 1/4 is 25%, 1/8 is half of 1/4 so it's half of 25% or 12.5%)
3/8 is 3 times that.
You should try doing this question in your head.

#225: To form a 4-person dance team, the order doesn't matter so you use choose method.
8C2 times 11C2 = ( 8 x 7 ) / 2 times ( 11 x 10 ) / 2 = 1540.

#226: Let x be Matt's new test average. From given you can set up the following equation:
(4x [the sum of the scores of the other 4 tests not including 60] + 60) / 5 = x - 5
4x + 60 = 5x - 25, x = 85

#227: If the dimension of the larger blocks is evenly divided by the smaller blocks, then it's a pretty
straightforward question. For example, how many 1 x 1 x 1 unit cubes could fit into a 4 x 4 x 4 larger cube.
(4/1) times (4/1) times (4/1) = 64 unit cubes.

This question is tricky because the dimensions don't line up nicely so you need to change the positions
of the smaller rectangular prisms in the later stages.
It's too difficult to draw out the solutions so here is the explanation.
Break the large rectangular prism into two parts, 9 x 5 x 5 and 1 x 5 x 5.
For the first part, you'll have (9/3) x ( 5/1) x (5/1) or 75 smaller rectangular prisms nicely fit in the large
rectangular prism.
For the 1 x 5 x 5 rectangular prism, you break it into 1 x 3 x 5 and 1 x 2 x 5.
For the first one, you can have (5/3) x (5/1) x (1/1) -- just multiply the integer quotient, or 1 x 5 x 1 =
5extra 3 x 1 x 1 rectangular prisms.
For the second one, you have have (1/1) x (2/1) x (5/3) or 2 extra rectangular prisms.
total comes up to 75 +5 + 2 = 82

#228: The middle three index cards are of the same length so the triangle is an equilateral triangle.
each angle is 60 degrees. The structure rests on a flat, level surface so if you make the two angles next to
the bottom angle of the triangle a and b, you will have the two angles a + b = 180 - 60 = 120.
Angle x and y each belongs to an isosceles triangle so x = 180 - 2a and y = 180 - 2b.
x + y = 180 - 2a + 180 - 2b = 360 - 2 (a + b) = 360 -240 = 120.

#229: (22 x 10- total distance- minus 4 x 16) / 6 = 26 mi/h

#230: 2/3 of the boys = 3/5 of the girls Times both sides by 15 and you have 10 b = 9 g
b = 9 and g = 10, 2/3 of 9 = 6 and 3/5 of 10 = 6 (19 - 6 - 6) / 19 = 7/ 19

2011-2012 Mathcounts warm-up 15

2011-11-26T12:28:00.000-05:00

#211: If the data doesn't overlap, you have the least possible number of students who have o pets, grandparents, or babies living in their homes.
1500 - 1500 (1/4 + 1/5 + 1/3) = 325

#212: This question is pattern recognition. The sequence looks like this: 5, 7, 1, 3, 1, 3...
After the first two terms you have 1, 3, 1, 3 ...repeating. 100 - 2 = 98
98 divided by 2 leave no remainder so the answer is "3".

#213:

#214:

#215: Use Pythagorean to get the hypotenuse:(3/2)² + (20/3)² = h²
9/4 + 400/9 = (81 + 1600)/36 = h², h = 41/6
The perimeter is (3/2) + (20/3) + (41/6) = (9 + 40 + 41)/6 = 90/6 = 15 = x
The area of the triangle = (3/2)(20/3)(1/2) = 5 = y
x² - y² = 225 - 25 = 200

#216: To find x²y²z² in the expansion of
(x + y + z)⁶, you do 6!/2!2!2! = 90
Same way as you find how many ways to arrange banana (6!/3!2!, 3 and 2 being the
repeating letters.)
Here are the explanations from Math Forum.

#217: If you count 0 as multiple of any digit, there are 9 ( from 1 to 9) numbers.
For unit digit 1, there is 1.
For unit digit 2, or other prime numbers, there are two numbers, which is 1 and itself so 4 x 2 = 8 numbers.
(2, 3, 5, 7, 4 prime numbers)
For unit digit 4, there are 1, 2, 4, three numbers.
Fro unit digit 6, there are 1, 2, 3 and 6, four numbers.
For unit digit 8, there are 1, 2, 4, and 8, four numbers.
For unit digit 9, there are 1, 3, and 9, three numbers.
Add them up and the answer is (9 + 1 + 8 + 3 + 4 + 4 + 3)/ 90 (two digit numbers from 10 - 99)
= 32/90 = 16/45

#218: Working backward, you'll see the value of each coin to the others looks like this:
1f = 4e = 16d = 64c = 256b = 1024a
To have least amount of coins, you use the larger value coins as many as possible.
2012 = (1 x 1024) + (3 x 256 ) + ( 3 x 64) + (1 x 16) + ( 3 x 4)
The lease number of coins are 1 + 3 + 3 + 1 + 3 = 11 coins

#219: For m < n and m + n is even, you need m and n both to be either even or both to be odd.
There are 50 even numbers and 50 odd numbers in that list.
50C2 + 50C2 = 2450 {Only 1 way that gives you m < n after you choose two even from the list of
even numbers.)
C
#220: CE = 7

2011-2012 Mathcounts workout 7

2011-11-22T22:10:00.000-05:00

#201: To get the surface area you do 2 x ( width x length + length x height + width x height) = 666 so width x length + length x height + width x height = 333
Volume = width x length x height = 720 , 720 = 24 x 5 x 3²
From the surface area you know that two of the dimensions have to be odd numbers to have
even + even + odd = odd.
Try (5, 3, 48), (15,3, 16) or (5, 9, 16) and you'll see (15, 3, 16) works. The longest length is 16.

#202: (14 x 81 - 10 x 75) divided by 4 = 96

#203:The answer is 0.73.

#204: To find the midpoint, add the two end points and divided the result by 2.
The answer is (4, -6).

#205: T is odd and S is prime and E > T. Write down the prime numbers. There
are 2, 3, 5 and 7. ATE + AT = ROYS You can see that R = 1 and A is 9 and O is
because two numbers adding together the most carry over is 1.T can't be 7 because
that leave only 8 for E and it doesn't work. 5 and 2 encounter similar problems.
So only 934 + 93 = 1027 works.
A + T + E + O = 9 + 3 + 4 + 0 = 16

#206: 2 students = 2/5% so total students are 2 divided by 2.5% =
2 divided by 0.025 = 80 students.

#207: AP : PB = 1: 2 so P is 1/3 x (5-0) = 5/3 away from 5 or 10/3
for the x value and 1/3 times 2 + 0 = 2/3 for the y value.
Plug (10/3, 2/3) to y = kx and you get k = 1/5

#208: Half circumference of the larger circle plus the whole circumference
of the smaller circle = 1/2 of 2r∏ + r∏ = 2∏r.

#209: Let Arlene started with x dollars. The equation would look like this if you keep finding the remaining fraction. {{[(3/5 of x - 4) times 1/2] - 5} times 2/5}- 6 = 2. Working backward and you get x = 90 dollars.

#210: The first time Judith, Mark and Kelly all received a gift is on the 49th day.
The least common denominator of 6, 5 and 4 is 60. ( 365 - 49 ) divided by 60 = 5. 26...
5 days plus the first one (the 49th day) is 6 days.

2011-2012 Mathcounts warm-up 14

2011-11-22T14:47:00.000-05:00

#191: Prime factorize 2940 first. 2940 = 3 x 5 x 7 ² x 2 ²
Keep only the perfect square factors and you have (7 ²)¹ x (2 ²)¹.
Use the exponent + 1 and multiply together and you have ( 1 + 1 ) ( 1 + 1 ) = 4
The four square factors are 1, 2², 7² and 2² x 7².

#192: All the same and you have (1, 1, 1), (5,5, 5) and (10, 10, 10)
three scores.
Two the same and one different and you have (1, 1, 5), (1, 1, 10),
(5, 5, 1),(5, 5,10), (10, 10, 1) and (10, 10, 5) six scores.
All different and you have (1, 5, 10) one score, so the answer is 10 different
total scores.

#193: For exponent questions, make the base the same so you can compare the
exponent and solve for the variable or make the exponents the same so you can
compare the base to see which is larger.
This one is the first case. 2¹⁸ = (2^-2)^x
18 = -2x so x = -9

#194: Diameter is a line so if you increase the line, which is 1-D, the volume, which is 3-D will be increased to 2³ = 8 times.
8 times 5 = 40 cubic meters of air.

#195: Use 30-60-90 angle degree ratio and you can get the value of CM + CN = 18 cm

#196: Let the side length of J = x. Since the triangle in the middle is a right triangle, using Pythagorean
theory and you have 7² - 2² = x² = 45 and that is the area of the square J.

#197: Kathy ate 1/8 of the jelly beans.
Sue ate one-fifth of the rest so she ate 7/8 (what's left after Kathy ate 1/8) x 1/5 = 7/40
Pat ate twice as many as Kathy and Sue combined so he ate 2 x ( 1/8 + 7/40) = 24/40
Drew ate the rest so he ate 1 - 1/8 - 7/40 - 24/40 = 4/40
/Drew : Pat = 4/40 : 24/40 = 1/6

#198: The three digits could be (1, 2, 6), (1, 3, 5), or (2, 3, 4). In these cases, order matters so each
three-digit group has 6 arranges (or 3! = 6) so 6 x 3 =18.
You can also have (0, 1, 8), (0, 2, 7), (0, 3, 6), (0, 4, 5). In these cases 0 could not be at the hundredth
digit so you only have 4 arrangements for each three-digit group ( or 2 x 2 x 1= 4) so 4 times 4 = 16
18 + 16 = 34 positive integers.

#199: Find the slope first. (y1 - y2)/ (x1 - x2) = - 3/2
The equation of the line is y = -3/2 of x + b. Plug in any point to get b = 3.
The equation of the line is thus y = (-3/2) times x + 3. The y-intercept is when x = 0 so y = 3
(same as the value of b). The answer is (0, 3).

#200: According to the given information, the known numbers look like this:
___ ___ 29___ ___ 35___ ___ 39___ ___. To make the mean the greatest you want to have a many
larger number as possible without violating the numbers above.
12, 29, 29, 35, 35, 35, 39, 39, 39, 52, 52 The mean is 396 / 11 = 36

2011-2012 Mathcounts warm-up 13

2011-11-19T17:20:00.000-05:00

#181: There are 4 horizontal lines and 4 vertical lines. If you choose two of any horizontal lines, combining with two of any vertical lines, you will have a rectangle.
There are 4C2 times 4C2 = 6 x 6 = 36 rectangles.
Of all these, there are 3² + 2² + 1² = 14 squares.
14/36 = 7/18

#182: Two congruent right triangles, each with hypotenuse 13 and two lengths of 5 and 12 forms this isosceles triangle. The area is 24 x 5 divided by 2 = 60 cm²

#183: 1/2 + 1/4 + 1/6 + 1/8 = 7/8 + 1/6 = (42 + 8)/ 48 = 50/48 = 25/24

Some mental math tricks:
1/2 + 1/4 = 3/4
1/2 + 1/4 + 1/8 = 7/8
1/2 + 1/4 + 1/8 + 1/16 = 15/16
.
.
.
1/a + 1/b = (a + b)/ab
1/2 + 1/3 = 5/6
1/3 + 1/5 = 8/15
.
.
.
c/a + d/b = (bc + ad)/ ab

#184: You can use Simon's Favorite Factoring Trick to solve this problem.
2a² - ab - b² = a² - ab + a² - b²= a (a - b) + (a - b) (a + b) = (a - b) ( a + a + b) = (a - b) (2a + b)

Or you can just divide the equation.

#185: rate x time = distance or rate = distance divided by time
Total distance is 3 x 20 + 2 x 30 = 120 and total time is 2 + 3 = 5 hours so the average speed is
120 divided by 5 = 24mi/h

#186: You can't have "0" in the thousandth digit so there are 9 numbers for the thousandth digit,
9 (this time, "0" is allowed.) for the hundredth digit, 8 remaining for the tenth digit and 7 for the unit digit.
9 x 9 x 8 x 7 = 4536

#187: The surface area of circular cylinder is 2∏ r² + 2∏r times height.
Since the height is twice the diameter or 4 times the radius, substitute the radius and you get
2 ∏ r² + 2∏ r times 4 r = 10 ∏ r² = 160∏. r = 4.
The volume of the cylinder is ∏ r² times 8r =8∏ r³ = 256∏.

#188: If Ben has x animals, Jerry has x + 10.
2x + 4 (x + 10) = 2x + 4x + 40 = 220. 6x = 180 so x = 30.
Jerry has x + 10 or 40 toy animals.

#189: There are total 10 marbles but 5 are the same blue color and the other 5 are the same green color,
Thus there are 10! / 5! x 5! = 252 ways to arrange them.
Only two ways render the marbles in alternate colors, which is BRBR... or RBRB...
2/252 = 1/126

#190: If the volume of an enlarged cube is 8 times the volume of the original cube, it means that
each side is (∛8) or twice the original side. The percentage increase is (2 -1)/1 = 1 = 100%

2011-2012 Mathcounts workout 6

2011-11-17T09:10:00.001-05:00

#171: Let n be the number of pennies Mollie has, then n < 100.
n divided by 5 leaves a remainder of 3 so the unit digit of n is either 3 or 8. (Why?)
n divided by 7 leaves a remainder of 1, combine with the previous condition, n is 43 or 78.
n is divisible by 3 so 78.

#172: From the center o, the triangle OAB is an equilateral triangle because OA = OB (radius)
= AB (given) so angle C is 30 degrees. (half of the central angle.)
Formulas for angles in circles

#173: It's always easier to work this kind of questions backward and reverse the operation.
15.54 divided by 1.05 divided by 0.8 = 18.50

#174: The length of the longer side is 4 x 2 (diameter of the larger circle) + 4 x 2 ( the sum of the two diameters of the smaller circle) = 16
The shorter length is 4 x 2. The area of the shaded region is 8 x 16 (area of the rectangle) - 2 x 2²Π - 4²Π
= 52.6017 = 52.60

#175: P = 5, Q = 3, P + Q = 7 and P - Q = 2. Their sum is 17.

#176: Bob runs 80 yards and Rob runs 80√ 2 yards.
Since they both arrive at the same time and Rob runs longer distance, his speed is faster and is
directly proportional to the distance. ( R times T = D so R and D are directly related.)
80√ 2/80 = x / 8, x = 8 times √ 2 = 11.3137 = 11.3

#177: There are 5, 9, 10, 12 and n five numbers. If n < 5, 9 would be the median.
The sum would be 9 x 5 = 45, but 45 - ( 5 + 9 + 10 + 12) = 9 (doesn't work)

If n is larger or = to 5 or smaller or = to 9, then 9 is still the median and 45 - 36 = 9 works in this case.

If n is 10, then 10 is the median but 10 x 5 - 36 = 14 (doesn't work)

If n is larger than 14, then it works so there are two integers n that work.

#178: There are 200 students and 68 vote for Grizzly Bear, which means that
(200 - 68) = 132 don't vote for Grizzly Bear. 132/200 = 33/50

#179: Prime factorize 96 and you have 96 = 3¹ x 2⁵
Use the exponents to get how many factors and you have (1 + 1)times (5 + 1)
= 12 factors

#180: See the image, the answer is 80 unit square.
You can also use the shoestring method to get the same answer.

2011-2012 Mathcounts warm-up 11

2011-11-14T11:32:00.000-05:00

#151: Multiples of 6 and 7 work on each column, so the answer is 6 + 7 = 13

#152: Two dice so there are 6 x 6 = 36 ways to pair the numbers.
Of those, 1-2 ( so is 2-1), 2-3 ( so is 3-2), ... 5, 6 ( so is 6, 5) the positive difference between
the numbers rolled is 1.
(5 x 2) / 36 = 10/36 = 5/18

#153: A line that is perpendicular to the segment of another line has a slope that is negatively
reciprocal to the line segment. The slope of line AB is (4.9 - 2.9)/ [-8.1 - (-7.6)]= 2/-0.5 = - 4
The negative reciprocal is 1/4.
Slope of a line

#154: 1 can't be in the single digit or the tenth place of the two-digit number. (Why?)
5 can't be in any of the unit digit of the three numbers because 5 x even number = 0 and 5 times
odd numbers = 5 and there is no extra 5 or 0 that's included in the 6 numbers.
Using trial and error, you get 54 x 3 = 162

#155: 32 divided by 2 = 16, 16 x 1 = 16 (The points you won if you predicted right in each
of the math outcome.
16 divided by 2 = 8 (games) and 8 x 2 = 16
8 divided by 2 = 4 (games) and 4 x 4 = 16
4 divided by 2 = 2 (games) and 2 x 8 =16
Finally 2 divided by 2 = 1 (games) and 1 x 16 = 16
16 x 5 = 80

#156: Another dimensional change problem. When the circumference is tripled, the area is 3²
or 9 times the original area. (1D versus 2D). The original volume of the cylinder is 3 times that of the
cylinder, so the ratio of the volume of the cylinder to the volume of the cone is 3: 9 or 1: 3.

#157: See the image.The answer is 17 ways.

#158: ( x - y ) / ( z - y ) = -2 so x - y = -2z + 2y, x = 3y - 2z
.( x - z ) / ( y - z ) = ( 3y - 2z - z) / ( y - z ) = (3y - 3z ) / ( y - z ) = 3 ( y - z ) / ( y - z ) = 3

#159: The area of the donut shape with the hole is 8²∏ - 4² ∏ = 48∏.
The area of the square is 8²/ 2 = 64/2 = 32
The answer is 48∏ + 32.

#160: There are 10 competitors and if each plays each other player once, there will
be 10C2 or 10 x 9/ 2 = 45 games played, which means there will be 45 winners and 45
losers.
45 divided by 7 = 6... If there are 6 winners with a record of 7 or more wins,
there is 4 losers. If they lose every game they play, 4 x 9 = 36, which is not
enough.
However, if there are 5 winners and 5 losers, 5 x 9 = 45 games lost,which will work.

2011-2012 Mathcounts workout 5

2011-11-10T12:12:00.000-05:00

#141: 40% of 80 is 32 and 32% of 75 is 75 x 32/100 or 3/ 4 of 32 = 24
75% = 3/4 and 3/4 of (32 - 24) = 6

#142: For the three-digit area code, since you can't use "0" or "1", you have (10-2)-digits for
each slot, or 8 x 8 x 8 choices.
(8 x 10 x 10 - 4)x 8 x 10⁶ = 6368000000 = 6.368 x 10⁹

#143: The area of the circle is 2²∏ = 4∏
The area of the hexagon is √ 3 /4 times 2² times 6 = 6√ 3
Their difference is 2.1740 = 2.17.

#144: Let the price of the first bike be x dollars and the price of the second bike be y dollars.
1.3 x + 1.5 y = 1.45 (x + y), 1.3x + 1.5 y = 1.45 x + 1.45 y
0.05y = 0.15x Times 100 both sides and you have 5y = 15x so x/y = 1/3.

#145: You want the hundredth place value to be as close as possible to get the least value so
it's either 6 and 7 or 2 and 3.
723 - 684 = 39
346 - 287 = 59 so 39 is the answer

#146: This one is a special right triangle question. The ratio of a 30-60-90 degree special right
triangle is 1-√ 3 -2. Since ∠ ACB is congruent to ∠CAD {BC and AD are parallel, so the
alternate interior angles are the same degrees.
AD/ CD = √ 3/1 = AD/40 so AD = 40√ 3
The area of the rectangle is 40 times 40√ 3 =2771.28 = 2771.3

#147: 1 Q(itch) + 2 G(ritches) = 20 --- equation 1
2Q(itches) + 1G(ritch) = 25 --- equation 2
Add up the two equations and you have 3Q(itches) + 3G(itches) = 45 so
1Qitch + 1Gitch = 15---equation 3
Using equation 1 to subtract equation 3 and you have 1Gitch = 5 points

#148: Convert the time to hour and you have 18 minutes = 18/60 hour = 9/30 hour
and 8 minutes = 8/60 hour or 4/30 hour.
Let x hour be the time that Sam gets to work on time and we can set up the equation
such as this: 30( x + 9/30) = 45 (x - 4/30)
Times each side by 2 to get rid of the fractional denominator.
60x + 18 = 90x -12, 30 = 30x so x = 1
Plug in either equation to get the distance. 30 (1 + 9/30) = 30 x (39/30) = 39 miles.

#149: A set with 10 elements have 2¹⁰ or 1024 subsets. [Each element is either included or
not included in the set, thus 2 to the 10th power.
Of all, half will be with odd number of elements in the subsets and half will be with even numbers
of elements in the subset so the answer is 1024/2 = 512.

#150: See the image.The answer is 35 degrees.

#150:

2011-2012 Mathcounts warm-up 10

2011-11-09T09:34:00.000-05:00

#131: Pair A: 6 x 4 = 24
Pair B: 16 + 8 = 24
Pair C: 36 -12 = 24, The common result is "24".

#132: 18 coins and three are quarters so there are 18 - 3 = 15 left for nickels and dimes.
There are twice as many dimes as nickels so there are 2/ (2+1) or 2/3 of the total coins that
are dimes. 2/3 of 15 is 10 so there are 10 dimes and 5 nickels, and the total comes up to $1.25.

#133: The side length of each square is 12 but after overlapping the two congruent squares
the new side is 20, which means that the over lapping part is (12 + 12) - 20 = 4
(Same as Venn Diagram two circles.)
12 - 4 = 8 so the side of the right side of the rectangle is 8 and the length is 12 (given) so
the area of the shaded triangle is half of the area of the rectangle or (8 x 12) / 2 = 48 square units.

#134: When the question asks for a product of a set of integers = another number, it's mostly a
factoring question.
84 = 1 x 84 (discard, since the number is greater than 1) = 2 x 42 = 3 x 28 = 4 x 12 = 6 x 8
= 2 x 2 x 12 (discard, since it asks for distinct) = 3 x 4 x 4 (discard, not distinct) = 3 x 4 x 7
3 + 4 + 7 = 14 or 6 + 8 = 14 [ The closer the set of numbers, the lower the sum.]

#135: The artist can draw 20 pictures in 240 minutes or 4 hours.
If she can draw 3 times as fast, she can draw 3 x 20 = 60 pictures in 4 hours.
For 6 hours, she can draw 6/4 times 60 or 3/2 times 60 = 90 pictures.

#136: A circle has 360 degrees. There are 12 hours on a clock face so one hour equals
360/12 or 30 degrees and half an hour = 15 degrees.
90 - 15 = 75 degrees.

Another way to look at this question is to look for how many degrees gained for each minute.
there are 60 minutes in an hour so 360/60 = 6, which means every minute the minute hand will
move 6 degrees, and every minutes the hourly hand will move 30/60 or 1/2 of a degree.
Combining the two and you get every minute the hourly and minute hand will be apart 5.5 degrees.
3: 30 = 3 x 60 + 30 or 210 minutes
210 x 5.5 = 1155 - 360 x 3 = 75

The second solution works best when the time is not as straightforward as o'clock sharp or
half hour, quarter hour...

#137: Imagine you can squash the block, you'll see the front and back, top and bottom, and
side and side are of the same area.
There are (10 x 2 ) = 20 front and back, ( 7 x 2 ) = 14 side and side, and (7 x 2) = 14
top and bottom.
Add the extra "2" block faces that make the U shape and you got 50.

#138: The slope of the two points P (1, 3) and Q (17, 43) is (43 - 3)/ (17-1) = 40/16 = 5/2
40 divided by 5 = 8 so there are 8 equal integer spaces between 3 and 43, which means "7"
integer solutions.
You can also use 16 divided by 2 = 8 to get the same result. [8 equal spaces so 7 integer
solutions between 1 and 17.

#139: This one calls for systematic reasoning.
20
= 1 + 2 + 17

= 1 + 3 + 16

= 1 + 4 + 15

= 1 + 5 + 14

= 1 + 6 + 13

= 1 + 7 + 12

= 1 + 8 + 11

= 1 + 9 + 10

= 2 + 3 + 15

= 2 + 4 + 14

= 2 + 5 + 13

= 2 + 6 + 12

= 2 + 7 + 11

= 2 + 8 + 10

=3 + 4 + 13

= 3 + 5 + 12

= 3 + 6 + 11

= 3 + 7 + 10

= 3 + 8 + 9

= 4 + 5 + 11

= 4 + 6 + 10

= 4 + 7 + 9

= 5 + 6 + 9

= 5 + 7 + 8

Total there are 24 distinct ways without numbers repeating.

#140: 7 x 6 (exit a different doors so after you choose the first door, now there is
only 6 doors left to choose) = 42 doors

2011-2012 Mathcounts warm-up 9

2011-11-08T14:16:00.000-05:00

#121: The answer is H.

#122: This is a ratio proportion question. The ratio of above ground pole to below ground is 4 to 1, which means the length of the pole that is above ground is 4/(4+1) or 4/5 of the total length.
4/5 of 30 = 24 feet.

#123: Let the original three-digit integer be ab0 or 100a + 10b + 0.
The larger three-digit number is ba0 or 100b + 10a + 0.
Their difference is (100b + 10a + 0) - (100a + 10b + 0) = 90b - 90a = 90(b - a) = 540
so b - a = 6
Since a + b = 12 (given) and b - a = 6, we know that a = 3 and b = 9.
The answer is 390.

#124: Each die has six sides, so the answer is 6 x 6 x 6 (there are three dice total) = 216 outcomes.

#125: Since this is an isosceles triangle, angle F = angle G = 1/2 (180 - 50 [angle A]) = 65 degrees
Angle BCE is the supplementary angle of angle ACB and the line BC // DE // FG so
angle ACB = Angle AGF = 65
BEC = 180 - 65 = 115 degrees

#126: You can just use the 2 and 1 as the length of the two legs of each of the four congruent triangle since
ratio and proportion stays constant in this case.
The percentage of the larger square which is shaded is [(1x2)divided by 2 times 4] / (3 x 3) = 4/9 or 44%

#127: The probability that Christoph will get two As is 1/4 times 1/4 = 1/16 (0.25 = 1/4)

#128: AD is 10 (given) and C is the midpoint (given) so CD = 10/2 = 5.
The area of the triangle CDE is 30, so (CE x 5)/2 = 30, CE = 12.
BC = 2/3 of CE = 8. Using Pythagorean theory and you get AB² = 5² [the length of AC²] + 8² [the length of BC²] = √ 89

#129: First member took 1, second 3, third 5, etc..., so this is sum of the first n "ODD" natural numbers, which = n² = 8² = 64. 64 + 5 + 3 = 72 milk balls.

#130: When the point P(3, 2) is reflected across the y-axis, the new point is (-3, 2).
Translating up 4 units and the answer is (-3, 6).

Some links on mastering fraction : Beginning level

2011-11-07T08:49:00.001-05:00

Level 1: From PBS Melvin's Make a Match

Level 2: From PBS Mission Magnetite

Level 2: Mathsplat

2011-2012 Mathcounts workout 4

2011-11-05T13:06:00.000-04:00

#111: Similar shapes share the same ratio and proportion, so 8/2 - radius of the larger cone versus smaller cone) = 18/ height of the smaller cone . The answer is 4.5 in.

#112: See solutions on the left images.

The answer is 72.

#113: Since the first two letters must be either "K" or "W", you have 2 choices for the first letter.
The other 3 letters can be any letter except "K" or "W" so there are (26 -2)³ choices. [There are 26 letters total. You got 24 letters left if you leave k and w.
The answer is 2 x (26 -2)³ = 27, 648.

#114: In an hour, a hare will run 1 x 60 = 60 meters.
A tortoise will crawl 1 cm time 60 = 60 cm a minute,and 60 x 60 = 3600 cm an hour = 36 meters an hour.
60 - 36 = 24 meters.

#115: A day has 24 x 60 = 1440 minutes. 1440 x 150 = 216000
Expressed in scientific notation with three significant digits gives you 2.16 x 10⁵.

#116: The length of the rectangle is increased by 75% so along with the original 100% now it's (100% + 75%) = 175% = 1.75.
The width of the rectangle is increased by 25% so along with the original 100% now it's (100% + 25%) = 1255 = 1.25
1.75 x 1.25 = 2.1875
For percentage increase, you find the positive difference and divided by the original number.
(2.1875 -1) / 1 = 1.1875 = 118.75%

#117: If the new-added solutions is 10x milliliters, there will be x acid solutions.
50% of 50 is 25 acid solution for the original liquid.
(25 + x) / 50 + 10x = 20% or 1/5
Use cross multiplication and you have 50 + 10x = 125 + 5x; x = 15
The scientist will have 50 + 10x or 50 + 10 x 15 = 200 milliliters of misture

#118: Use ratio relationship and you have 3.5 in./ 42 mi. = 2.75 in./ x mi.
Again, use cross multiplication to get 42 x 2.75 divided by 3.5 = 33

#119: The height of the first bounce is 5 ft so after another 5 bounce, it will be the sixth bounce of the ball.
Each bounce the ball reaches 80% or 4/5 of it's original height.
5 x (0.8)⁵ = 1.6384 The answer is 1.6.

#120: To find percentage, you use part / total. To get the left over area, you use the area of the rectangle minus the area of the circle.
[8.5 times 11 - (4.25²π)] / 8.5 x 11 = 0.39310 = 39.3%

2011-2012 Mathcounts warm-up 8

2011-11-03T10:15:00.000-04:00

#101: A pasture with an area of 25 hectares is enclosed using 2 km of fencing. Another pasture uses
8 km of fencing and the two fenced areas are similar. Since perimeter is 1-D dimension or a line and
the area used to enclose the hectares are 2-D, the ratio is 1: (8/2)² = 1 :16.
16 x 25 = 400 hectares

#102: From 101 to 200, starting from the unit digit that is 3, you have 103, 113, ...193, ten numbers.
From tenth digit that is 3, there are 130, 131, ...139 another ten numbers so total 20 numbers.
From 201 and above, if you start from the unit digit that is 3, there are 203, 213, ...243 five numbers.
For the tenth digit to be 3, there are 230, 231, 232 ... to 239, another ten numbers.
Total there are 35 numbers.

# 103: 3x + 3 = 2x + 7; x = 7 - 3 = 4

#104: Every hour Tim drove 30 miles and Kim drove 3 times as long as time so 3 x 40 (mi/h) = 120.
Tim : Kim = 30 : 120 = 1 : 4

Solution I:
Let Tim drove 1 times x miles, then Kim drove 4x miles.
1x + 4x = 5x = 225; x = 45 miles

Solution: II:
Time drove 1/ (1+4) times the total 225 miles = 45 miles

#105: xz : bd = 4 : 18 = 2 : 9
If the line ratio is 2: 9, the area ratio is 2² to 9² = 4 : 81
The area of triangle ABC = 243 so the area of the triangle wxy can be solved by using ratio
(243/81) = (wxy/4); wxy = 12 cm²
(4 times WY) divided by 2 = 12; WY = 6 cm

#106: Add the two end points and divided the result by 2 will give you midpoint.
(-2 + 3)/ 2 = 1/2 = 0.5; (4 + -3)/2 = 1/2 = 0.5. The answer is (0.5, 0.5)

#107: 21 - (9 + 4 -2) = 21 - 11 = 10 students

#108: The range is the positive difference between the largest and the smallest numbers so
16 - 4 = 12-- range
(4 + 5 + 7 + 7 + 8 + 8 + 8 + 9 + 16) divided by 9 = 8 -- mean
12 - 8 = 4

#109: Learn those Pythagorean Triples since they come up at competition math a lot.
This one is 6 times the 3-4-5 right triangle so the diagonal is 6 x 5 = 30. In a rectangle,
both diagonals are congruent so the answer is 30 x 2 = 60 m

#110: Rate times Time = Distance so Distance divided by Time = Rate
117 divided by 2.25 = 117 divided by 9/4 = 117 times 4/9 = 52 mi/hour

2011-2012 Mathcounts warm-up 7

2011-10-31T20:32:00.002-04:00

#91: 130 x 7 (rent 7 days) + 50 = 960 dollars

#92: The circumference of a circle is Dπ or 2πr and the area of a circle is πr². For them to be equal, 2r = r² so r = 2

#93: 4 x 3 = 12 cm since the fly is magnified to three times its original size.

#94: 1 - (1/5)[the probability of choosing a red token] - (1/3)[the probability of choosing a blue token] = 1 - (8/15) = 7/15

#95: The smallest three-digit positive square number is 10² or 100 and the largest three-digit positive square number is 31² or 961 so there are
31 - 10 + 1 = 22 numbers

#96: There are 21 customers so the median is (21 + 1) / 2 = 11, the 11th customer.
Start with the single digit and work up in orders and you have 5, 6, 8, 10, 10, 14, 18,18, 22, 22,and 25. The 11th is the number 25.

#97: Since all the diameters, and 1, 2, 3, 4 is the smallest integer length.
(1 + 2 + 3) x 2 + 4 = 16

#98: Learn the divisibility rules:

Divisibility test from mathleague

vectorkids: divisibility rules

If the sum of the digits of a number is divisible by 3, the number is divisible by 3.
6 + 4 + 5 + D = 15 + D; Since 15 is divisible by 3, as long as D is divisible by 3, the number is divisible by 3. Thus D could be 0, 3, 6, 9 and their sum is 18.

#99: 1/m + 1/2m = 6; The least common multiple of m and 2m is "2m" so 2/2m + 1/2m = 3/2m = 6;
3 = 12m so m = 1/4.

#100: One side of the square is s so it's area is s². The base of one small triangle is s and the height is b so the area is sb/2.

There are 4 small triangles so the total areas are 4 times (sb/2) = 2sb

The area of the star is thus s² - 2ab

Oct. 28th, 2011 Mathcounts beginning intermediate group lesson test

2011-10-29T08:36:00.001-04:00

Update:

Mental Math: All exceed a score of 20, one over 30.

Word problems: Only one got perfect; however, all are too slow.

These are not hard problems, so try harder and think smart.

1. A triangle with an area of 80 has a height of 5. What is the area of a similar triangle with a height of 10?

2. How many multiples of 9 are there between 60 and 200?

3. A circle was cut into 8 congruent slices and 3 of those were eaten, what is the degree of the central angle?

4. A cookie recipe requires 4 cups of flour to make 6 dozen cookies. How many cups of flour do you need to make 9 dozen cookies?

6. Mary paid $85 dollars for a dress after 40% off the original price, plus an additional 20% off the sale price and a $5-off coupon. What was the original price if she didn’t have to pay sales tax?

7. What is the degree measure of an interior angle of a regular octagon?

8. A bicyclist wants to make a 360-mile trip on his two-wheel bicycle. He has a spare wheel which is used to replace either of the other two wheels. Suppose each of the three wheels is to have the same mileage for the trip. How many miles should each wheel travel?

9. The first fifteen multiples of 3 are 3, 6, 9, 12…45. What is the sum of these multiples of 3?

10. How many line segments can you draw if there are 6 points on a circle?

Mental Math:

1. 11 x 45

2. 0.99 x 7

3. 20 x 3 x 45

4. 1/5 + 1/7

5. 12.5 % of 128

2011-2012 Mathcounts workout 3

2011-10-29T08:20:00.000-04:00

#81: If 2 out of 3 students attend the Festival, then 1 out of 3 students will not attend.
1140 divided by 3 = 380.

#82: 18 shots is equivalent to 60% or 3/5 of the total attempts. 3/5 of x (total attempts) = 18
x = 18 divided by 3/5 = 18 times 5/3 = 30 attempts.
Practice solving similar questions in your head.

#83: 8 (4 divided by 0.5) x 8 (4 divided by 0.5) x 6 (3 divided by 0.5) = 288 cm²

#84: If the original cost is x, after 8% sales tax it would cost x times 1.08 (100% + 8% = 108% or 1.08). After another 20% tip for the waiter, she would pay
x times 1.08 times 1.2 (100% + 20% = 1.2) = 16.20

The original cost is 16.20 divided by 1.08 divided by 1.2 = 12.5 dollars

#85: The key thing you need to consider is the 30 cents, since you know you have to have 2 pennies and 1 dime (10 cents) is much lighter than 2 nickels.
Again, for 30 cents, it's lighter if you use 3 dimes instead of 1 quarter and 1 nickel, so the answer is 4 x 2.268 (dimes) + 2 x 2.5 (pennies) = 14.072g

#86: See the explanations on the left. The answer is 66 cm²

#87: Use proportion to solve this question. 2670/5340 = 3185/x
Cross multiply and you get x = 6370

#88: (233- 25) divided by 8 = 26

#89: From points A and B, you get the two ys' distance is 8; from A and C you get the two xs' distance is 15. These form two legs of the triangle so the area of the triangle is (8 x 15) divided by 2 = 60 unit²

#90: Trapezoids ABCD and WXYZ are similar so all their corresponding sides are in the same proportion. AS/XW = 9/6 = 1.5 so 3x - 6 = 1.5 (x + 1)

Both sides times 2 and you have 6x -12 = 3x +3; 3 x = 15 so x = 5
AD = 3x -6 = 9

2011-2012 Mathcounts Sequences Stretch

2011-10-25T22:32:00.001-04:00

#283: The 100th term of the arithmetic is 3(the first term) + (100-1) x 8 = 795

#284: The sum of any arithmetic sequence is average times the terms (how many numbers in that sequence)
The average is (3 + 795) /2 = 399. Since there are 100 numbers. 100 x 399 = 39,900.

#285: The first term in that geometric sequence is 729 and the ratio is 243/729 or 1/3 so the 10th term of the geometric sequence is 729 x 1/3^(10-1) =729 x 1/3⁹ = 1/27

#286: From 1st to 18th, there are (18 - 1) spaces. (8.25-4) / 17 = 0.25
8.25 + (35 - 18) x 0.25 = 12.5

#287: p, 2p +6, and 5p -12 are arithmetic sequence so (2p +6) - p = 5p -12 - (2p +6)
p +6 = 3 p -18; 2p = 24 so p = 12; The first term is 12 and the second term is 30 so the difference is 18. The 4th term is 12 + (4-1) x 18 = 66

#288: n, n + 3, and 2n +6 are geometric sequence so (n+3)² = n (2n +6);
n² + 6n + 9 = 2n² + 6n; n² = 9 so n = 3
3, 6, 12, 24, so 24

#289: (83-17) /(9-3) = 11, which is the space. 17 - 2 x 11 = -5

#290: (81/24) / r^{5th term - 2nd term} = r³; (81/24)^1/3 = 1.5 = the common ratio, so the first term is 24/1.5 = 16

#291: 24 x 2 = 48. Any arithmetic sequence, the middle number is 1/2 of the sum of the two numbers right beside it.

#292: There are 8 cells, or 2³ at the end of the 2nd hour, so at the end of 3rd hour there will be 2⁴.... At the end of the 8th hour there will be 2⁹ = 512 cells.

2011-2012 Mathcounts warm-up 6

2011-10-25T09:34:00.000-04:00

#71: When forming a team, order doesn't matter so use 6C3 = (6 x 5 x 4)/(3 x 2 x 1) = 20

#72: The volume of any type of prism/cylinder is base times height. In this case, the height is 9 am and the volume is 144πcm³, the area of the base is thus 144 /9 = 16πcm².

#73: 7 x 8 = 56 so total you have 56 slices. The fraction of the leftover pizza slices is (56 - 41)/56 = 15/56

#74: Let the number be x. √ 2x -3 = 3; Square both sides and you have 2x - 3 = 9; x = 6

#75: Suppose Bailey has x marbles and Kaylee has y marbles. According to the given information, you can set up the two equations.
x + 2 = 2 ( y - 2) ; x = 2y - 6 --- equation 1
3(x -3) = y + 3 ; y = 3x - 12 --- equation 2
Substitute x = 2y - 6 (equation 1) to equation 2 and you have y = 3 (2y -6) - 12 = 6y - 18 - 12
-5y = -30 so y = 6; Plug in equation 1 and you get x = 6 as well, so the answer is "0".

#76: Use different color pens to mark the horizontal and vertical lines and you'll see some of the same length appears twice.

#77: The sum of the first 5 rounds of golf is 5 x 98 = 490 strokes; After the next 3 rounds, if the average is 92, the sum would be now [5 + 3 (the next three rounds)] x 92 =736
(8 x 92 - 5 x 98) / 3 = 82

#78: There are 20 space between -8 and -3 and each is [-3 - (-8)] / 20 or 0.25
There are 10 space between b and a so their positive difference is 2.5; however, the question asks the value of b - a so the answer is -2 and 1/2 as a mixed number.

#79: The first coat, after 20% discount, now cost 80 x (100% - 20%) = 80 x 4/5 = 64 dollars
Using the same method, the second coat now cost 100 x 7/10 x 9/10 = 63 dollars (Do this mentally!!)
The positive difference is 1 dollar.

#80: The key information is y = x. From that you can figure out the three vertices of the triangle are
(1, 1), (5, 5) and (5, 1). The base is thus (5-1) = 4 and the height is also (5 -1) = 4 so the area is (4 x 4)/ 2 = 8

2011-2012 Mathcounts Warm-Up 5

2011-10-24T17:34:00.003-04:00

#61: This is the same question asking you what is the sum of the first consecutive 11 natural numbers.
It's an arithmetic sequence so to get the sum, you use average times the terms (how many numbers)
The average is (1 + 11) divided by 2 and there are 11 numbers, so the answer is 6 x 11 = 66
Review the sum of the first natural number here.

#62: The diameter of each circle is 8 so the radius is 8/2 = 4. If you connect the three centers of the circle, it will form an equilateral triangle with a side length 4.
Use either angle ratio or Pythagorean theorem to get the the area of an equilateral triangle given a side, the answer is √ 3 / 4 times 4² = 4√ 3

#63: In this geometry sequence, the ratio is -4, so the next number is -256.

#64: The area of the largest circle is 81∏, which means that the radius is of the largest circle is 9.
The diameter of the largest circle is thus 9 x 2 = 18, and the diameter of each of the smaller circle is
(18 divided by 3) or 6 units; its circumference is thus 6∏.

#65: 2x + 3 = 4, 12x + 18 is exactly 6 times the previous equation (2x +3) so the answer is 6 x 4 = 24.

#66: If you count the non-overlapping white regions, there are 5 regions that the two lines divide, so five.

#67: The mode is the year that appears the most so it's 1974.
There are 32 family members so the median is the average of the 16th and the 17th, which are both 1968.
Their positive difference is 1974 - 1968 = 6

#68: Each time 6 players will be on the field and there are total 8 players, which means that each play will be on the field 6/8 or 3/4 of the time so 3/4 of 48 is 36 minutes.

#69: Every minutes the dog will gain 2 feet so 150 feet divided by 2 = 75 leaps.

#70: Let x be the largest number and y be the smallest number.
x + 4 x 80 (largest number plus the sum of the remaining 4 number that has the mean of 80) = 4 x 90 + y
320 + x = 360 + y
x - y = 40

2011-2012 Mathcounts Workout 2

2011-10-23T07:24:00.000-04:00

#51: 10/61 = 0.1639344... = 16.39 % (To figure out the percentage, you do part/ total.)

#52: 15 miles/per hour (given) = 15 x 5280 (feet)/ 60 minutes ; Simplify and you get 1320 feet/per second.

#53: Malcolm will get 2 - 1.64 = 36 cents; The least number of coins would be 25 cents, 10 cents and one penny, so 3 coins.

#54: Roll four times and the third roll is a 3; so it's
7 x 7 x 1( the number 3) x 7[7 is the other number that is not 3] / 8 x 8 x 8 x 8 = 343/4096

#55: Area of a rectangle is width x length = 15 x width = 195; width = 13 (Do this mentally!)

#56: Watch this video first if you don't know how to solve Venn Diagram two circle problems.

12 + 11 - 6 + 3 = 20 students

What about Venn Diagram three circles? Watch this video.

#57: The area of a trapezoid is (base1 + base2) times height and then divided by 2
The height is half of the shorter base (given), so 8/2 = 4; (8 + 11) x 4 / 2 = 2 x 19 = 38 cm ²

#58: AC = 2x + 5 + 6x -1 = 8x + 4 = 36; x = 4
AB = 2x +5 = 13

#59: The first 5 counting numbers are 1, 2, 3, 4, 5. There are two ways, which is 1 + 2 and 2 + 1, that the sum of the two numbers will be 3.

With replacement, the answer is 2(ways)/(5 x 5) = 2/25

#60: 3p(potato salad) + 4c(coleslaw) =10.75
1p(potato salad) + 2c(coleslaw) = 4.75
Add up the two equation and you get 4p + 6c = 15.50
Divide that equation by two and you get 2p + 3c = 7.75 dollars

2011-2012 Mathcounts Warm-up 4

2011-10-22T14:51:00.001-04:00

#41: It's easy to get what the first column, second row is : a triangle.
Same with the fourth column, second row: a circle.
For column two, row three. It appears at first you can put a triangle, but it won't work since the last row has already a circle.
Thus, it should be a circle and then a triangle for the second column.
Once that's solved, you can get the shaded space quite easily.
The answer is a triangle.

#42: Two angles are supplement if they add up to 180 degrees.
To get the third angle, you use [180 - 7 - 97 = 180 - (7 + 97)] ; to get the supplement of the third angle, you do 180 - the third angle, thus the easiest way to do this problem is just add up the two interior angle
7 + 97 = 104 degrees

#43: 7/4 = 1.75 so the positive difference between 3.75 and 7/4 = 2 [Do this mentally!!]

#44: Parts/ Total = 2/ 10 = 1/5

#45: The probability of choosing the first girl is 8/20, or 2/5; the probability of choosing the second girl is 7/19. The answer is 14/95

Other questions to ponder:
1. What is the probability of choosing two different boys?
2. What is the probability of choosing one boy and one girl?
3. What is the probability of choosing two different girls and one boy?

#46: First, know these definitions.
3 sides = triangle
4 sides = square/rectangle//parallelogram/rhombus/trapezoid and other quadrilaterals
5 sides = pentagon
6 sides = hexagon
7 sides = heptagon

8 sides= octagon

9 sides = nonagon

10 sides = decagon

12 sides = dodecagon

There are at least two ways to find the interior angle of a regular polygon.

method I: Find how many non-overlapping triangles you can make, then times 180. Afterwords divided by the sides of the polygon.
With a little practice you'll see each time you can make (n-sides -2) non-overlapping triangles.
So for pentagon, (5-2) x 180 / 5 = 108 degrees

method II: No matter how many sides a polygon has, it's interior will add up to 360 degrees. (Can you prove it?) Thus, 360 divided by the sides will give you one exterior angle. 180 - that exterior = one interior.
For pentagon, 180 - (360 / 5) = 180 - 72 = 108 degrees.

#47: There are 4 more nickels than dimes and two more quarters than nickels. Both refer to nickels so let there be x nickels and you have (x - 4 ) dimes and (2 + x) quarters.
There are 37 coins total so x + (x - 4) + (2 + x) = 3x -2 = 37; x = 13
13 + 2 = 15 quarters

#48: There are at least 2 ways to solve this problem.
Method I: Let there be "x" ounces of flavored syrup added to the original mixture and you have
(25% of 12) / 12 + x = 1/10 ; 3/(12 + x) = 1/10 so x = 18 ounces flavored syrup

Method II: 25% of 13 is 1/4 of 12 = 3 ounces of active ingredient, once the flavored syrup add to the original mixture, the 3 ounces turn into 10% of the original mixture, which means that it now is 3 divided by 10% or 30 ounces.
30 - 12 = 18 ounces

#49: A trillion have 12 zeros so 12.5 x 10¹² = 1.25 x 10¹³ so 13.

#50: One compensation plan is $1000 all at once.

The second plan:
1st day 1 dollars
2nd day 2 dollars or the sum is (1+2) = 3 = 2² - 1
3rd day 4 dollars or the sum is (1 + 2 + 4 ) = 7 = 2³ - 1
4th day 8 dollars or the sum is (1 + 2 + 4 + 8) 15 = 2⁴ -1
.
10th day sum = 210 - 1 = 1024 -1 = 1023
The positive difference is 1023 - 1000 = 23 dollars

Mathcounts Beginning intermediate Group

2011-10-22T13:16:00.001-04:00

2011-2012 Mathcounts Handbook We have covered till workout 2. Make sure you know how to do all the previous warm-up, workout questions.

Homework:
A: review the two tests, all the problems we cover so far from our previous lessons.
B. Mental math a score of at least 20.

First test

1. supplementary angle of 37 degrees

2. complimentary angle of 43 degrees

3. One side of a heptagon (or septagon) is 7 cm, what is its perimeter?

4. One interior angle of a nonagon.

5. It takes a mechanic 5 hours to fix 2 car, how many hours will it take him to fix 3 cars?

Second Test:

1. The sum of 7 consecutive numbers is 84, what is the smallest number?

2. Andy has 3 T-shirts, 2 hats and 5 pairs of shoes. How many days can he wear a different outfit before it has to be repeated?

3. List all the prime two digit numbers with a unit digit that ends in “1”.

4. After 25% off and another 20% off plus a 5 dollar coupon, a dress costs $61 dollars. What is the original cost of the dress?

5. How many multiples of 9 are there between 200 and 400?

6. What is one interior angle of a nonagon?

7. What is the supplementary angle of the smaller angle formed by the hands of a clock that displays a time of 10 o’clock?

8. Two hexagons are similar. One has a side of 3 cm and an area of 32 cm square. The side of the other similar hexagon is 3 times the smaller one. What is the area of the hexagon?

9. The third of an arithmetic sequence is 30 and the 6^th is 75, what s the 10^th number?

10. There are 24 numbers evenly spaced on a clock face. What number is opposite 6?

Problems we cover in class

1. The mean of a set of 13 numbers is 21. When four consecutive integers are added to the set, the mean is now 27. Find the largest integer that was added to the original set.

2. Find the units' digit of the sum 17³²⁵ + 23¹⁰⁸

3. Arrange the four numbers 2 4, 6, 8. What is the sum of all those arrangement?

Arrange the four numbers 1, 3, 5, 9. What is the sum of all those arrangement?

Other problems:

1. What is the sum of all of the four-digit positive integers that can be written with the digits 1, 2, 3 and 4 if each digit must be used exactly once in each four-digit positive integer? (03 chapter sprint # 30)

2. Camy made a list of every possible distinct five-digit positive even integer that can be formed using each of the digits 1, 3, 4, 5 and 9 exactly once in each integer. What is the sum of the integers on Camy's list? (04 chapter # 29)

3. What is the average (mean) of all 5-digit numbers that can be formed by using each of the digits 1, 3, 5, 7, and 8 exactly once? (2005 AMC-10 B)

4. A group of n people are selecting entrees at a restaurant. There are 5 entries in all. What is the smallest possible n such that we can be sure that at least 3 people have dishes from the same category?

Countdown Round -- at the end

2011-2012 Mathcounts Workout 1

2011-10-17T22:16:00.000-04:00

#21: Do this in your head. An hour later is 11:30 but an hour is 60 minutes not 55 minutes, which means the actual time is 11: 25. In 35 minutes it will be noon.

#22: Use calculator to add up the sum and you have $27.41. 27.41 - 17.36 = 9.78 = 3.49 + 6.29
6.29 - 3.49 = 2.80 dollars

#23: The unit cubes that have no faces painted red are the one hidden inside each of the outer two layers.
(5 - 2) (5-2)(5-2) = 27 unit cubes

#24: 14 yards = 14 x 3 = 42 feet and 6 yards = 6 x 3 = 18 feet

(42 / 6) -1 = 6 and (18/6) - 1 = 2; these are the posts that are not on the corner.
6x 2 ( two opposite length) + 2 x 2 (two opposite width) + 4 (corners) = 20 posts

#25: If the hundreds digit is 1, you have 101 and 110, two ways.
If the hundreds digit is 2, you have 211, 202 or 220, three ways.
If the hundreds digit is 3, you have 312, 321, 330, 303, four ways.
.
.
.
If the hundreds digit is 9, you have 909, 990, 981, 918, 927, 972, 963, 936, 945, 954, 10 ways.
2 + 3 + 4 +... + 10 = 1 + 2 + 3 + 4 +...10 - 1 = 55 - 1 = 54

#26: 4 (recommend)/ 7(total) = x / 1421; Cross multiply and you get x = 812
Make sure to simplify 7 with 1421 first.

#27: If length is 3x, width is x (why?) 3x² = 48, so x = 4 (width) and length = 12 ; √ 12² + 4²
= 12.6491 about 12.65

#28: 15 dozens is 3 times 5 dozen cookies, so Amy needs 3 x 4 = 12 cups of flour.

#29: Length: width: height = 3 : 2 : 1 (given). Let height be x inches long, then width would be 2x inches and length would be 3x inches long.
Volume = L x W x H = 3 x times 2 x times x = 6 x ³ = 48 in³
so x = 2; Length = 3 x = 6

#30: Dollar divided by how many units (cans) = unit price (single can cost)
(7.68 + 4.4) divided by 24 = 0.5033333..
So 51 cents.

2011-2012 Mathcounts Warm-up 3

2011-10-17T07:50:00.001-04:00

#31: A hexagon has 6 sides so if each side measures 6 cm, the perimeter is 6 x 6 =36 cm.

#32: How to find the median value from Math is Fun.
There are 5 numbers in this question, the median number is the third bar if you line them up in order.
The answer is 4.

#33: The least two-digit prime is 11 and the largest is 97.
11 x 97 = 1067 (Do this in your head!)

#34: This one is ratio and proportion question. If it takes the mechanic x hours to install carburetors the equation would look like this:
4 hours/ x hours = 3 cars / 5 cars, 20 = 3x, x = 20/3 hours
Convert it to minutes and you have 20/3 times 60 = 400 minutes

#35: There are total 2 + 3 + 4 + 3 = 12 cars and 3 of them are green.
The probability that the car is not green is (12 -3) / 12 = 9/12 = 3/4

#36: One yard = 3 feet and 1 feet = 12 inches
Convert 15 yards to inches, you have 15 x 3 x 12 inch long ribbon, which is evenly distributed to 20 craft projects so the maximum length of ribbon each can require is:

15 x 3 x 12 / 20 = 27 (Simplify first. Don't do busy work!!)

#37: Order of Operations question: Work from left to right for multiplication and division.
2 divided by 4 x 8 = 1/2 times 8 = 4

#38: Two angles are complementary when they add up to 90 degrees; two angles are supplementary when they add up to 180 degrees.

There are 12 hours in a clock and a clock (a circle) has 360 degrees so each hour is 360/12 = 30 degrees.

4 (o'clock) x 30 = 120 degrees so the measure of the supplement of of the smaller angle formed by the hands of a clock that displays 4 o'clock is 360 - 120 = 240 degrees.

#39: This one is tricky. First learn how to set up the correct equation; later you can practice divided by a fraction mentally.
A lot of students use what's given to solve this question, which is not correct.
If an item is 45% off, you are not paying 45% of the item cost but (100%-45%), or 55% of what the item costs.
Thus, the equation should look like this:
Let the item costs x dollars, (100%-45%) (100%-20%) times x - 5 (coupon) = 50
55% times 80% times x = 55
Simplify and you have 11/20 times 4/5 times x = 55
11/25 times x = 55, x = 55 times (25/11) simplify again and you have 5 x 25 = 125 dollars

#40: If you know 0.125 = 1/8, then this question would be very easy to solve.
Let the number be "a" , according to the given information, you can set up the equation as follows:
a times x = a divided by 0.125 = a divided by (1/8) = 8 times a, x = 8

2011-2012 Mathcounts Warm-up 2

2011-10-11T12:36:00.121-04:00

#11: Make sure you know the order of operations.

Please Excuse            Parenthesis Exponents (Radicals)
My Dear                    Multiplication/Division left to right
Aunt Sally                  Addition/Subtraction left to right

2 x (3+4) = 2 x 7 = 14; 2 x 3 + 4 = 10
The difference is 4.

#12: Use easier example to help you reason this question out.
If you have 3 pairs of shoes and 2 shirts, how many days can you wear different combinations for outfits?
Let there be A, B, and C (three pairs of shoes) and a, b (two shirts).
You'll see you can wear 3 x 2 (not 3+2) = 6 different combinations for outfits.

For this question, 5 (shirts) x 3 (pairs of pants) x 2 (ties) x 4 (pairs of shoes) = 120 different uniforms

#13: A circle has 360 degrees. Two slices were eaten so there are (12 -2)= 10 slices not eaten.
10/12 = 5/6 ; 5/6 times 360 = 300 degrees.

#14: Let width be x and the length will be 2x + 6.
2x + 6 = 78; x = (78-6) /2 = 36 ft

#15: Ratios and proportions questions regularly pop up on both competition math and standardized tests.
Make sure you know how to approach them. They are very straightforward.

Solution 1: A : B = 2 : 3 Let A be 2x and B be 3x (same ratio proportion)
A + B = 2500 (given) = 2x + 3x = 5x; x = 2500/ 5 = 500
B = 3x = 3 x 500 = 1500 dollars.

Solution 2: A : B = 2 : 3, which means every 5 parts (2 + 3 = 5), A would have 2 parts and B have 3 parts,
so a is 2/5 of the total and B is 3/5 of the total.
2500 times (3/5) = 1500 dollars [Make sure to simplify first. Don't do busy work.]

#16: There are 3, 5, 7, 11, 13, 17, 19, which totals to 7 odd primes.
7/20 is 35% [ both numerator and denominator times 5 will give you the percent desired since 20 x 5 = 100.]

The others similar to remember are: 10 x 10 = 100; 25 x 4 = 100; 12.5 x 8 = 100; 125 x 8 = 1000.

Use these to your advantage to solve the percent problems fast.

#17: Order of operations question.
5 x ( 11 + 4 divided by 4) = 5 x ( 11 + 1) = 5 x 12 = 60 [Do this in your head!]

#18: Two similar polygons, the perimeter ratio is 1-D so if the perimeter of the smaller decagon is 76, the larger one would just be 76 x 2 = 152 cm

Now if the area of the decagon is 76 cm 2 , the area of the other similar decagon with a double side length would then be 76 x 2 2 or 76 x 4 = 304 cm 2.
Why?? Make sure you totally understand this.

#19: This is a trick question.
On the 1st day, you paddle 3 miles north and back 2 miles south, so you only progress 1 mile.
On the 2nd day, you paddle 1 + 3 = 4 mile north and back 2 miles so 2 miles.
On the 3rd day, you paddle 2 + 3 = 5 miles north and back 2 miles so 3 miles.
                    .
                    .
On the 47th day, you paddle 46 + 3 = 49 miles north and back 2 miles so 47 miles.
On the 48th day, you paddle 47 + 3 = 50 miles so you are done.

The answer is 48. [Yeah! Free at last.]

#20: ABCDEF is a regular hexagon and BC is parallel to AD. Connect all the opposite vertexes and you'll break the hexagon into 6 equilateral triangles, and AD = 2BC = 2 x 6 = 12.

2011-2012 Mathcounts Warm-up 1

2011-10-11T12:14:00.003-04:00

Practice answering all the questions in your head. This sheet is pretty easy.

#1: There are 12 months in a year, so 35 years have 35 x 12 or do it mentally, 35 x (10 + 2) = 350 + 70 = 420 months

#2: Plug in the x and y values and you'll get a = 3

#3: This one is very important. There are numerous questions on "how many numbers (terms)".

Use easier examples to help you reason it out.

From 1 to 9, there are 9 numbers, not (9-1) or 8 numbers. The range just shows how many space in between the two numbers, so you need to + 1 in the end.

For this question, from 21 to 157, there are (157 - 21) divided by 2 [every two numbers are odd, thus divided by 2] = 68 space so 68 +1 = 69 numbers

#4: Each side of the square is the same length, so x - 2 = 3 and x = 5.

#5: 20% of the grapes are green, which means that 80% of the grapes, or 4/5 are not green. 4/5 of 50 is 40 (grapes).

#6:The radius of the circle is 4 so the diameter is 2 x 4 = 8. The area of the square is 8 x 8 = 64 square ft.

#7: Solution I : The area of the triangle is (base times height) divided by 2. We know the height and area so substitute and we get (base x 10) / 2 = 120 so base = 24

The base of the other similar triangle with double the height is thus 24 x 2 (In a similar triangle, each side is the same ratio) = 48

The area of the triangle is (20 x 48) / 2 = 480 mm²

Solution 2: In any similar polygon, if one dimension changes to x ratio, the area will change to x² of the original area (Area is 2-D, so squared.)

In this case, the height is doubled so the area will change to 2² or 4 times the original area.

120 x 4 = 480 mm²

If it's asking you the volume relationship, then it's x³ (Volume is 3-D, so cubed.)

There are just too many similar questions to count for this concept.

Make sure you totally understand how it works and try more problems to secure your knowledge.

#8: There are 3 geometry books and 7 algebra books, so total 10 math books.

The probability of the first book is math book is 10/16 or 5/8.

One thing to keep in mind is that for any probability questions, the answers always fall between 0 and 1 inclusive, 0 being "never happens" or impossible and 1 being "always happens".

If your answer is not at that range, it's wrong.

#9: Volume of a rectangular prism is width times height times length so plug in the given information, you have 5 x 4 times height = 120 so height is 6 inches.

#10: Prime factorize 96 and you get 96 = 3 x 2 ⁵ so the greatest prime factor is 3.

2011-2012 Mathcouts season starts Yeah!!

2011-10-11T11:12:00.001-04:00

Here you can find 2011-2012 Mathcounts handbook problems.

On my next post, I'll start writing down the solutions that students and I have come up with at our meets.

Please refrain yourself from just copying down the answers. That just wastes your time and lots of people's good intentions.

Also, don't give up too readily. Try at least twice. Google the concepts involved and if still not getting it, ask for hints.

Have fun problem solving!!

Take care and cheers, Mrs. Lin

Algebra questions without using the variables

2011-06-10T18:28:00.000-04:00

#1: There are 20 horses and chickens at Old Macdonald's farm. Together there are 58 legs. How many horses and how many chickens?

Solutions:

#1: Using algebra, you have

H + C = 20---equation 1 and

4H +2C = 58---equation 2

To get rid of one variable you can times equation 1 by 2 to get rid of Chicken or times 4 to get rid of horses.

Times 2 and you have 2 H + 2 C = 40 (every term needs to be multiplied by 2)---equation 3

4 H + 2C = 58---equation 2

Using equation 2 - equation 3 and yo get 2H = 18 so H = 9 and from there, solve for C = 11

#2: Without using algebra, you can make all the animals be chickens first. In that case, you'll have 20 x 2 = 40 legs. Since you have 58 legs, you need to get rid of some chickens and bring in more horses.

You gain 2 legs by every transaction (-2 + 4 = 2). (58 - 40) / 2 = 9 so 9 horses and 11 chickens.

Rate, time, and distance question

2011-05-27T07:36:00.000-04:00

# 1: Esther drove to work in the morning at an average speed of 45 miles per hour. She returned home in the evening along the same route and averaged 30 miles per hour. If Esther spent a total of one hour commuting to and from work, how many miles did Esther drive to work in the morning?

#1 Solutions:

A: Let t be the time it took Esther to drive to work and (1-t ) be the time it took him to return home.

Since rate times time = distance, we can set the equation as 45 t = 30 (1- t), 75t = 30 so t = 30/75 or 2/5

2/5 (time) times 45 = 18 miles, which is the answer.

B: The rate ratio between driving to work and returning home is 45 to 30 or 3 to 2.

Since rate and time are inversely related (rt = d), the time ratio between the two is 2 to 3.

45 time (2/5) = 18 miles

C: Let D be the distance from Esther's home to work.

D/45 + D/30 = 1 (hour); Times 90 to get rid of the denominator and you have 2D + 3D = 90

D = 18 miles

A Skill for the 21st Century: Problem Solving

2011-03-22T21:30:00.001-04:00

A Skill for the 21st Century: Problem Solving by Richard Rusczyk, founder of "Art of Problem Solving".

Mathcounts tricky Algebra questions

2011-03-05T21:02:00.000-05:00

#24 1993 National Sprint: Bottle A contains more Diet Coke than Bottle B. Now do the following:
a. Pour from Bottle A into B as much Diet Coke as B already contains.
b. Pour from B into A as much Diet Coke as A now contains.
c. Pour from A into B as much Diet Coke as B now contains.
Both bottles now have 64 ounces. How many more ounces were in A than in B at the beginning?

#30: 1993 National Sprint:Auggie spent all of his money in 5 stores. In each store, he spent $1 more than one-half of what he had when he went in. How many dollars did Auggie have when he entered the first store?

Solution:

#24: Let A contains x ounces and B contains y ounces and x > y (given).

After first pouring, A has (x - y) ounces left and B has 2y ounces (double the original amount)

After second pouring, A has ( 2x - 2y)(double the amount) ounces and B has (3y - x) ounces left.

After third pouring, A has (3x - 5y) ounces left and B has (6y - 2x) (double the amount)

3x - 5y = 64 times 2 for each terms 6x - 10y = 128 ----equation 3

6y - 2x = 64 times 3 for each terms 18y - 6x = 192 ---- equation 4

equation 3 + equation 4 and you have 8y = 320 and y = 40 ; Plug in any equation and you get x = 88

88 - 40 = 48 ounces

#30: Let Auggie had x dollars at the beginning. At the first store, he would spent 1 + (x/2) = (2+x)/2 and would have x - (2 +x)/2 = (x-2)/2 left

At the second store, he would spend 1 + (x-2)/4 and would have (x-2)/2 - 1 - (x-2)/4 or (x-6)/4 left

At the third store, he would spend 1 + (x-6)/8 and would have (x-14)/8 left

It looks like there's a pattern. At the fourth store, he would spend (x-30)/16

and at the 5th store he would spent (x-62)/32 = 0 so x - 62 = 0 and x = 62

2011-03-04T13:58:00.002-05:00

Continental Math Question: Three boys, AI, Bill and Chuck, play a game with the understanding that the loser is to double the number of marbles of each of the other two. After 3 games, each had lost once (AI first, then Bill, then Chuck) and each ends up with 24 marbles. How many marbles did AI start with?

#27 1993 Mathcounts National Sprint: Three players play 4 rounds of a game and end with a tie score of 80. A different player wins each of the final three rounds. When a player wins, s/he doubles his or her points, and the two losers each must subtract the amount the winner gains from their scores. What was the highest score at the end of round 1?

Solution:

Solving this question backwards so you start with C.

Continental Math question:

A----------B----------C

24 24 24 after third game

12 12 48 after second game

6 42 24 after first game

39 21 12

#27: Make a List and work it out backwards. In my case, I assume C wins first, then B then A.

A---------B---------C

80 80 80 after third game

40 120 120 after second game

100 60 180 after first game.

190 150 90

The answer is 190.

Some fraction questions

2011-02-26T20:57:00.000-05:00

#3 1993 National Target: A worm was measured in a laboratory and its growth was charted. At the end of one week, its length has increased by one-half. At the end of the second week, its length had increased by one-third from the previous week, the third week by one-fourth from the previous week, and so on. After how many weeks was the worm 20 times as long as when it was first measured?

Partition Questions

2011-02-23T15:45:00.000-05:00

#24 2001 Sate Sprint Round: The number 4 can be written as a sum of one or more natural numbers in exactly five ways: 4, 3+1, 2 + 1 + 1, 2 +2 and 1 + 1 + 1 + 1; and so 4 is said to have five partitions. What is the number of partitions for the number 7?

Extra: Try partition the number 5 and the number 8.

Solution:

#24: You can solve this problem using the same technique as counting coins:

Counting coins questions

7 6 5 4 3 2 1

1 1 way

1 1 2 ways ( 5 + 2 or 5 + 1 + 1)

1 1 1 way

1 1 2 ways ( 4 + 2 + 1 or 4 + 1 + 1 + 1)

2 0 1 1 ways

1 2 3 ways ( 3 + 2 + 2, 3 + 2 + 1 + 1 and 3 + 1 + 1 + 1 + 1)

3 4 ways (2 + 2 + 2 + 1, 2 + 2 + 3 ones, 2 + 5 ones and 7 ones.)

Total 15 ways.

The partitions of 5 are listed below (There are 7 ways total.):

5 4 3 2 1

1 1 way

1 1 2 ways (3 + 2 and 3 + 1 + 1)

2 3 ways (2 + 2 + 1, 2 + 1 + 1 + 1 and 1 + 1 + 1 + 1 + 1)

There are 22 ways to partition the number 8.

Some counting and probability questions on dot grids

2011-02-22T13:14:00.001-05:00

#5 1993 Mathcounts National Target : Find the probability that four randomly selected points on the geoboard below will be the vertices of a square? Express your answer as a common fraction.

#5 2004 AMC 10A: A set of three points is chosen randomly from the grid shown. Each three-point
set has the same probability of being chosen. What is the probability that the points lie on the same straight line?

Solution:

#5 National Target: There are 16C4 = (16 x 15 x 14 x 13)/ 4 x 3 x 2 = 1820 ways to select 4 points on the geoboard.

There are 3 x 3 = 9 one by one squares and 2 x 2 = 4 two by two squares and 1 x 1 = 1 three by three squares. (Do you see the pattern?)

There are 4 other squares that have side length of √ 2
and 2 other larger squares that have side length of √ 5.

9 + 4 + 1 + 4 + 2 = 20 and 20/1820 = 1/91

#5 AMC-10A: There are 9C3 = (9 x 8 x 7) / 3 x 2 x 1 = 84 ways to chose the three dots and 8 of the lines connecting the three dots will form straight lines. (Three vertical, three horizontal and two diagonals.) so

8/84 = 2/21

Percent practices for young mathletes

2011-02-21T16:53:00.000-05:00

From Quizville : Solve some percent problems mentally.

Type in the number, find the desired percentage and check your answers through virtual representations.

Mission Magnetite from PBS

Fraction/Percent/Decimal Jeopardy from Qia

Geometry questions: equilateral triangles

2011-02-18T19:17:00.000-05:00

#1: P is the interior of equilateral triangle ABC, such that perpendicular segments from P to each of the sides of triangle ABC measure 2 inches, 9 inches and 13 inches. Find the number of square inches in the area of triangle ABC, and express your answer in simplest radical form.

#2: AMC 2007-B: Point P is inside equilateral ABC. Points Q, R, and S are the feet of the
perpendiculars from P to AB, BC, and CA, respectively. Given that PQ = 1,PR = 2, and PS = 3, what is AB ?

#3:

This is an equilateral triangle. If the side is "S", the length of the in-radius would be √ 3 /6 (or 1/3 of the height) and the length of the circum-radius would be
√ 3/3 (or 2/3 of the height).

You can use 30-60-90 degree special right triangle angle ratio to get the length of each side as well as the height.

Solution I: Let the side be "s" and break the triangle into three smaller triangles.

12S (base times height divided by 2) =

(√ 3 / 4) times S²

S = 16√ 3

Area of the triangle = 192√ 3

Solution II: Let the side be "s" and the height be "h", the area of the triangle is 24s = sh

(Omit the divided by 2 part on either side since it cancels each other out.)

h = 24, Using 30-60-90 degree angle ratio, you get 1/2 of s = 8√ 3 so s = 16√ 3

Area of the equilateral triangle = (24 times16√ 3 ) divided by 2 = 192 √ 3

#2: This one is similar to #1, the answer is 4√ 3

2007 Mathcouts State Sprint Geometry Questions

2011-02-17T19:38:00.001-05:00

#24: Each lateral edge of a regular hexagonal pyramid is 13 inches long. The altitude is 12 inches long. How many inches are in the perimeter of the hexagonal base?

#25: In equilateral triangle ABC, points M and N are the midpoints of AB and AC, respectively. Given that the area of triangle BPC is 18 square units, how many square units are in the area of triangle ABC?

#29: Right triangle POQ is similar to right triangle XYZ. QO = 4 and QP = 10. Given that
XY= √ 189, how many units are in the length of ZY ?

Solution:

#24: The height is 12 inches and the lateral edge of a regular hexagonal pyramid is 13 inches (given), which

means that the length from the center of the hexagonal base to each of the vertex is 5. (5-12-13 right triangle)

Thus each side is 5 inches (A regular hexagon consists of 6 equilateral triangles) so the perimeter is 6 x 5 = 30

inches

# 25: Solution I;

Triangle MPN is similar to triangle CPB. MN to BC = 1: 2 so their area ratio is 1: 4, which means that the area of

triangle MPN = 18/4 = 4.5.

Triangle MPN and MPB share

the same vertex M so their area

ratio is the same as NP to PB

= 1 : 2 so the area of triangle

MPB = 2 times 4.5 = 9

, the same goes with triangle NPC and CPB.

The area of the trapezoid is 3/4 of the area of triangle ABC so ABC = (9 + 9 + 18 + 4.5) divided by 3/4 = 54

Solution II: Draw a line connects the two midpoints M and N. Triangle MPN is similar to triangle CPB so the

height ratio of the two triangles is 1: 2. Thus AQ to PQ = 3 : 1 The area of the triangle ABC is 3 x 18 = 54

(Both triangle BPC and ABC share the same base so the area ratio is 3: 1)

#29: Triangle POQ is similar to triangle XYZ.QO = 4 and QP = 10 (given) so

OP = √ 100-16 = √ 84

OP/OQ = √ 84 /4 =√ 189/YZ

YZ = 6

2004 Mathcounts state sprint

2011-02-16T20:01:00.001-05:00

#19: The points (x, y) represented in this table lie on a straight line. The point (13, q) lies on the same line. What is the value of p + q? Express your answer as a decimal to the nearest tenth.

#24: The terms x, x + 2, x + 4, ..., x + 2n form an arithmetic sequence, with x an integer. If each term of the sequence is cubed, the sum of the cubes is - 1197. What is the value of n if n > 3?

#29: The graph shows six labeled points. How many distinct circles of radius 2 units are in the coordinate plane and pass through exactly two of the labeled points on this graph?

#30: Regions A, B, C, J and K represent ponds. Logs leave pond A and float down flumes (represented by arrows) to eventually end up in pond B or pond C. On leaving a pond, the logs are equally likely to use any available exit flume. Logs can only float in the direction the arrow is pointing. What is the probability that a log in pond A will end up in pond B? Express your answer as a common fraction.

Solution:

#19: Look at the table and you'll see each time x + 2, y would -3.

-5 to -14 is (-9), three times (-3) so p = 2 + 3 x 2 = 8

2 + 5.5 x 2 = 13 so q = -5 + (5.5) x (-3) = -21.5

p + q = -13.5

#24: The common difference in that arithmetic sequence is 2 and the cum of the cubes is -1197, which meant that the numbers are odd numbers. (cubes of odd numbers are odd and the sum of odd terms of odd numbers is still odd. )

(-5)³ + (-7)³ + (-9)³ = -1197 However, n is larger than 3 (given) so the sequence will look like this:

(-9)³+ (-7)³+ (-5)³+ (-3)³+ (-1)³ + (1)³ + (3)³ = -1197

-9 + 2n = 3; n = 6

#29: There are four circles with radius 2 using the four corner coordinates. (See the image below.)

There are 14 circles with radius 2 that you can make. ( See the image below.) Altogether 18 circles.

#29:

#30: You need to solve this questions by cases.

Case I: A - J - B : 1/3 times 1 times 1/2 = 1/6

Case II: A-K-B: 1/3 times 1/3 = 1/9

Add the two cases up and you have 5/18

Weird but delicous math question

2011-02-12T21:53:00.000-05:00

#4 1993 Mathcounts National Team Round:The teacher whispers positive integer A to Anna, B to Brett, and C to Chris. The students don't know one another's numbers but they do know that the sum of their numbers is 14. Anna says: "I know that Brett and Chris have different numbers." Then Brett says: "I already knew that all three of our numbers were different." Finally, Chris announces: "Now I know all three of our numbers." What is the product ABC?

Solution:

#4: For Anna to know that Brett and Chris have different numbers, she must have an odd number.

For Brett to say that he already know all three have different numbers, he not only must have an odd number, but also his number has to be larger or equal to 7.

Otherwise, A-B-C could be 1-1-12; 3-3-8; or 5-5-4. It would exceed 14 if you have 7-7-__.

If Brett has 7, then the numbers could be A-B-C = 1-7-6 ; 3-7-4 or 5-7-2.

If Brett has 9, then the numbers could be A-B-C = 1-9-4; or 3-9-2.

If Brett has 11, then then numbers could be A-B-C = 1-11-2.

From the above possibilities you know Chris has to have 6 for him to be sure he knows all the numbers.

So A-B-C = 1-7-6 and the product of ABC = 1 x 7 x 6 = 42

Problems that use Pascal's Triangle

2011-02-11T13:57:00.001-05:00

#3 2004 Mathcounts State Team: Coin A is tossed three times and coin B is tossed two times. What is the probability that more heads are tossed using coin A than using coin B? Express your answer as a common fraction.

#16 2008 AMC-10 B: Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls is odd? (Note that if no die is rolled, the sum is 0.)

# 12 2004 AMC-10 A : Henry's Hamburger Heaven off ers its hamburgers with the following condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, and onions. A customer can choose one, two, or three meat patties, and any collection of condiments. How many di erent kinds of hamburgers can be ordered?

#12 2007 AMC-10 A : Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible?

#10 2004 AMC-10 A: Coin A is flipped three times and coin B is flipped four times. What is the probability that the number of heads obtained from flipping the two fair coins is the same?

Solution:

#3: Use 1-2-1 (zero head, one head, two heads) for coin B cases and 1-3-3-1 (zero head, one head, two heads and three heads) for coin A cases.

For A to have more heads than B, there are three cases:

Case I: A- 1 head and B zero head. There are 3/8 times 1/4 = 3/32 probability.

Case II: A- two heads and B zero or one head. There are 3/8 times (1 + 2)/4 = 9/32 probability.

Case III: A- three heads. 1/8 and that's it. There is no other way that B will have more heads than A.

3/32 + 9/32 + 1/8 = 16/32 = 1/2

#16: Two coins so the probability is 1-2-1 (zero head, one head, two heads)

The probability of getting at least one head is 3/4 and no matter if you roll the dice once or twice and add up the sum, the probability of getting a sum of odd is 1/2 so 3/4 times 1/2 is 3/8.

#12: There are 8 condiments and you can choose either to put or not put on your hamburger so altogether you have 2⁸= 256 choices.

You have three choices for the meat patties so 256 x 3 = 768

#12: Solution I:

There are two tour guides A and B and both lead at least one tourist so there are

6C1 + 6C2 + 6C3 + 6C4 + 6C5 = 62 possible ways to group the tourists.

Solution II: 2⁶ - 2 because you can't have the cases that none belongs to A or B so 64 - 2 = 62.

#10: Use Pascal's Triangle 1-3-3-1 (zero, one, two, three) for A and 1-4-6-4-1 (zero, one, two, three, and four) for B.

1/8 times 1/16 + 3/8 times 4/16 + 3/8 times 6/16 + 1/8 times 4/16 = 35/128

Mathcounts Counting problems- ascending or descending numbers

2011-02-10T22:19:00.000-05:00

#6 1993 State Target:: A number is called increasing if each of its digits is greater than the digit immediately to its left, if there is one. How many numbers are there between 100 and 200?

#3: 2004 State Target: A license plate consists of two letters followed by two digits; for example, MP78. Neither the digits nor the letters may be repeated, and neither the letter O nor the digit 0 may be used. When reading from left to right, the letters must be in alphabetical order and the digits must be in increasing order.
How many different license plate combinations are possible?

#9 2006 AMC-12How many even three-digit integers have the property that their digits, read left to right, are in strictly increasing order?

Solution:

#6: 1 __ __. The first number is set so you can only start with at least 2. From 2 to 9 there are ( 9-2 ) + 1 = 8

numbers, so 8C2 = (8 x 7) / 2 = 28 because no matter what two numbers you choose, there is only one way to make it ascending.

You can also use the same method to get ascending numbers.

#3: __ __ (two letters except "o") __ __ (two numbers except "0")

There are 25 letters and 9 numbers remaining so 25C2 times 9C2 = 10800.

Again, there is only one way for the two letters to be in alphabetical order and one way for the two numbers to be in increasing order.

#9: __ __ 4 ; 3C2 = 3

__ __ 6; 5C2 = 10

__ __ 8; 7C2 = 21

3 + 10 + 21 = 34

Mathcounts Geometry Prolem:

2011-02-05T21:32:00.001-05:00

#7 1993 National Team: A water container with a circular bottom tapers linearly to a circular top with a radius one-half of the base. A mark halfway up the side says "200" gallons." How many gallons, to the nearest tenth, does the water container hold?

#7: Extend the circular top to make it a cone shape.

If the radius of the circular top is 1, then the bottom radius would be 2 and the radius for the midway circle would be 3/2. We are trying to find the ratio so any number will work since ratio stays constant.

The line ratio of midway radius to the bottom radius is 3/2 to 2 so the volume ratio would be (3/2)³ : 2³ = 27/8 : 8 or 27 : 64, which means the volume of the lower half water container is (64-27) or 37/64 of the total volume.

200 divided by 27/64 would give you the total volume of the cone; however, we need to minus the little top added cone, which is 1/8 of the total volume of the big cone. 200 divided by (37/64) x (7/8) = around 302.7 gallons

Mathcounts problems on exponents

2011-02-02T16:26:00.004-05:00

#27 1992 Chapter Sprint: In what digit does 2¹⁷ - 3 end?

# 8 1992 Chapter Team: What is the sum of the prime factors of 2¹⁶ -1?

#26 1992 State Sprint: Find the remainder when 2¹⁰⁰ is divided by 5.

#12 1992 National Sprint: When 2⁴⁰ -1 is written in standard notation,what digit is the unit's position?

#7 2007 State Target: When the expression (2¹)(2²)(2³)...(2⁹⁹)(2¹⁰⁰) is written as an integer, what is the product of the tens digit and the ones digit?

#2 1992 National Team: What are the last two digits of 2²²²?

#27 For this type of question, look for pattens and the unit digit.

2¹ - 3 = -1

2² - 3 = 1

2³ - 3 = 5

2⁴ -3 = 3

2⁵ - 3 = 9

2⁶ -3 = 1

2⁷ -3 = 5

2⁸ - 3 = 3

2¹⁷ - 3 = ?

17 - 1 = 16; 16/4 = 4...R = 0 so it would be the same as 2⁵ so the answer is 9.

Solution II: The pattern for 2 to the nth power is 2, 4, 8, 6.
Exponent 17 divided by 4 = ...R 1 so the unit digit ends in 2. Together with the tenth digit - 3, the answer is 9.

#8: Use difference of squares to solve this question a² - b² = (a + b) (a - b)

2¹⁶ - 1 = (2⁸ - 1)(2⁸ + 1) = (2⁴- 1)(2⁴ + 1)(2⁸ +1) = (2² - 1)(2² + 1)(2⁴ + 1) (2⁸ +1) = 3 x 5 x 17 x 257

The sum is 282. To check if the number 252 is a prime, you only need to check the prim smaller or equal to √ 257 = about 16.03

#26: This question is very similar to chapter 27. The answer is 1

#12: Again, this question is similar to chpater 27. The answer is 5.

#7: First of all, (2¹)(2²)(2³)...(2⁹⁹)(2¹⁰⁰) = 2^{1+2+3+...+99+100} = 2⁵⁰⁵⁰

The unit digit for the power of two repeats every 4 places, which is 2, 4, 8, 6, 2, 4, 8, 6...

The tenth digit for the power of two after 2 to the first power, repeat every 20 places, which are (02), [ 04, 08, 16, 32, 64, 28, 56, 12, 24, 48, 96, 92, 84, 68, 36, 72, 44, 88, 76, 52], 04, ---

5050 -1 = 5049; 5049 / 20 = ...R 9 so the tenth digit is 2 and the unit digit is 4

2 x 4 = 8, which is the answer.

#2: Use the same method as the previous question.
222 - 1 = 221; 221/20 = ...R 1 so the tenth digit is 0 and the unit digit is 4.

Some tricky counting/probability questions from Mathcounts/AMC

2011-02-01T11:24:00.002-05:00

#9: 2007 Chapter Team Round: Using the digits 2, 3, 4, 7, and 8, Carlos will form five-digit positive integers. Only the digit 2 can be used more than once in any of Carlos' five-digit integers. How many distinct five-digit positive integers are possible?

#21: 2003 AMC-10 A: Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. There are at least six of each of these three kinds of cookies on the tray. How many different assortments of six cookies can be selected?

#19 2001 AMC-10 Pat wants to buy four donuts from an ample supply of three types of donuts: glazed, chocolate, and powdered. How many different selections are possible?

#7: 1992 Mathcounts State Team: In how many different ways may 15 indistinguishable dimes be distributed among Susan, Frank, and Harold if each must receive at least one dime?

#24: 2002 AMC-10 A Tina randomly selects two distinct numbers from the set {1, 2, 3, 4, 5}, and Sergio randomly selects a number from the set {1, 2, ...10}. The probability that Sergio's number is larger than that sum of the two numbers chosen by Tina is?

#9: 2006 AMC-12 B: How many even three-digit integers have the property that their digits, read left to right, are in strictly increasing order?

Solution:

#9: If each number is different, you have 5! = 120 ways to achieve that.

If you have two 2s, you have 5! / 2! = 60 ways to arrange the numbers. However, you have 4C3 = 4 ways to choose the other three numbers from {3, 4, 7, 8} so 60 x 4 = 240 ways total.

If you have three 2s, you have 5!/3! = 20 ways to arrange the numbers. However, you have 4C2 = 6 ways to choose the other two numbers from { 3, 4, , 7, 8} so total 20 x 6 = 120 ways

If you have four 2s, you have 5!/4! = 5 ways to arrange the numbers. However, you have 4C1 = 4 ways to choose the other 1 number so 5 x 4 = 20 ways

There is only one way to make 22222.

Add all the combination up and you get 120 + 240 + 120 + 20 + 1 = 501 ways total.

#21: Solution I:

Case 1: 0-0-6 (All cookies are the same kind.) 3 ways

Case 2: 0-1-5 Five the same and one different. You have 3C1 x 2C1 = 6 ways.

Case 3: 0-2-4 Two of the same kind and the others of the other same type. You have 3C1 x 2C1 = 6 ways

Case 4: 0-3-3 Three-three. 3C1 x 2C1 = 6 However, CCC-OOO is the same as OOO-CCC so divided by 2 to get 3 ways.

Case 5: 1-2-3 You have 3! = 6 ways

Case 6: 2-2-2 You have 1 way.

The last case: 1-1-4 you have 3 ways.

So total you have 28 ways.

Solution II:

Add two dividers | | to break the cookies piles of different kinds.

11000000 means all peanut butter cookies.

00100100 means 2 chocolate chips, 2 oatmeal, and 2 peanut butter cookies.

10001000 means 3 oatmeal and three peanut butter cookies.

There are thus 8C2 = 28 ways to arrange the dividers so the answer is 28.

#19: This question is very similar to the previous one. 6C2 = 15

# 7: Since each must receive at least one dime, you first use 15 - 3 to get 12.

Now you can use two dividers to break the piles of indistinguishable dimes. 12 + 2 = 14

14C2 = 91 ways

#24: Tina chooses two distinct numbers from the set and since the order doesn't matter when you calculate the sum, she has 5C2 = 10 ways to do that.

For the denominator, there are 10 x 10 (10 numbers for Sergio to choose from) = 100

The possible sum for Tina is from 3 (1 + 2) to 9 (4 + 5). Further investigation you'll see there is one way to get 3, one to get 4, two ways to get 5 (2, 3 or 1, 4), two ways to get 6 (4, 2 or 5, 1), two ways to get 7

(4, 3 and 5, 2) one way to get 8 (5, 3) and one way to get 9 (4, 5)

For Sergio's number to be larger than Tina, starting from 1, you have 0-0-0-1-2-4-6-8-9-10, total 40 ways.

40/100 = 2/5

Solution II:

Tina's choices Sergio's choice larger than Tina

1-2 7

1-3 6

1-4 5

1-5 4

2-3 5

2-4 4

2-5 3

3-4 3

3-5 2

4-5 1

Sum total = 40

40 /100 = 2/5

#9: Let the three-digit numbers be abc. Since it's an even number and it's in increasing order, c can't be 0 or 2. (Why?)

If c is 4, there are (1, 2), (1, 3), (2, 3) or 3C2 = 3 ways to get the increasing number. (For any three-digit distinct number, there is only one way to be in increasing order; there is also only one way to be in decreasing order. )

If c is 6, there are 5C2 or 10 ways to be in increasing order.

If c is 8, there are 7C2 or 21 ways to be in increasing order.

Add them up and you get 3 + 10 + 21 = 34 such integers.

Mathcounts Strategy: Shoestring method of finding the area of any polygon

2011-01-31T15:37:00.003-05:00

#1: Find the area of a quadrilateral polygon given the four end points (3, 5), (11, 4), (7,0) and (9,8) in a Cartesian plane.

#2 2007 Chapter Target Round: A quadrilateral in the plane has vertices at (1,3), (1,1), (2, 1) and (2006, 2007). What is the area of the quadrilateral?

#3: Find the area of a polygon with coordinates (1, 1), (3, -1), ( 4, 4), and (0.3)

#4: What is the number of square units in the area of the pentagon whose vertices are (1, 1 ), ( 3, -1), (6, 2), (5, 6), and (2, 5)?

Solution I: Draw a rectangle and use the area of the rectangle minus the four triangles to get the area of the quadrilateral polygon.

Solution II: Using shoestring method. First, plug in the four points. Second, choose one starting point and list the other points in order and at the end, repeat the starting point. The answer is 33 square units.

#2 Answer: 3008 square units

#3: Answer: 10.5 square units

#4: Answer: 22 square units

AMC-10: 2010 AMC-10 A # 18 Probability

2011-01-29T21:05:00.000-05:00

#18 2010 AMC-10 A: Bernardo randomly picks 3 distinct numbers from the set {1, 2, 3, 4, 5, 6, 7, 8, 9} and arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set {1, 2, 3, 4, 5, 6, 7, 8|} and also arranges them in descending order to form a 3-digit number. What is the probability that Bernardo's number is larger than Silvia's number?

Solution I: Visit the AoPS Wiki.

Solution II: From Robert

There are 9C3 = 84 possible numbers for Bernardo and 8C3 = 56 possible numbers for Silvia, all of these numbers have one way to be ascending.

Let 1, 2, 3, ...84 be assigned to Bernardo's possible numbers in an ascending order.

Let 1, 2, 3, ... 56 be assigned to Silvia's possible numbers in an ascending order.

For the denominator, there are 84 x 56 total numbers of combination of triples.

If Bernardo gets one of the (84-56) = 28 possible numbers that are larger than all of Silvia's 56 numbers, he will win. This can happen in (84 -56) times 56 = 28 x 56 ways.

If Bernardo gets one of the numbers in Silvia's range (1-56 triples), the number of ways he can win is 56C2.

(28 x 56) + 56C2 / (84 x 56) = 37/56

Mathctounts Strategy: Finding the remaining fraction plus some tricky questions at the chapter level

2011-01-29T15:14:00.000-05:00

#1: A computer virus is eating disk space. During the first day it eats 1/2 of the disk.During the second day, it eats 1/3 of the remaining disk space. The third day it eats 1/4 of what still remains and the fourth day it eats 1/5 of what is left. What fraction of the original disk space remains intact?

#8 2007 School Target: Boar Ring gave a speech to an assembly of colleagues. After five minutes, half of the audience left; 10 minutes later a third of the remaining audience left. After twenty minutes, half of the remaining audience left, leaving only three people in the audience. How many people were in the audience at the beginning of Mr. Ring's speech?

#9. 2007 School Team: I have two one-quart jars; the first is filled with water, and the second is empty. I pour half of the water in the first jar into the second, then a third of the water in the second jar into the first, then a fourth of the water in the first jar into the second, then a fifth of the water in the second jar into the first, and so on. How much water is in the first jar after the 10th pour? Express your answer as a common fraction.

#8: In the circle show, with center at point A, rectangle AFGH has AF = 7 units and AH = 24 units. Points A, C, D, F, G and H are all in the same plane, and points C, D and G are on the circle. What is the length of segment CF?

#10: A basement has a 24-foot by 32 foot rectangular floor. The basement if flooded with water to a depth of 18 inches. Three pumps are used to pump the water out of the basement. Each pump will pump 8 gallons of water per minute. If a cube foot of water contains 7.5 gallons, how many minutes will it take to pump all of the water out of the basement using the three pumps?

Solution:

#1: The key strategy to this kind of question is to find the remaining fraction. On the first day, there is 1/2 left. On the second day, there is 2/3 of the remaining 1/2 left. On the third, there is 3/4 of what was left and on the fourth day, there is 4/5 of what was left on the previous day.

Multiplying this all out and you have 1/2 times 2/3 times 3/4 times 4/5 = 1/5 (cancel and simplify the calculation process.)

#8: Let there be x people n the audience at the beginning of Mr. Ring's speech.

Again, using the "find the remaining fraction" method, you will have 1/2 times 2/3 times 1/2 of x = 3

Simplify and you have 1/6 of x = 3 so x = 18 people

#9: This question is again utilizing the "finding the remaining fraction" strategy. Keep in mind you total have 1 quart of water to distribute between the two jars, so as long as you figure out what's left in one jar, the other jar would have (1- what's left) fraction left.

Jar 1 Jar 2

1st pull 1/2 1/2

2nd pull 2/3 1/3 (find the remaining fraction so 1/2 times 2/3 = 1/3)

3rd pull 1/2 (2/3 times 3/4) 1/2 (1/2 times 4/5 = 2/5 for the next pull)

4th pull 3/5 2/5

5th pull 1/2 1/2 Have you noticed that there is a pattern?

10th pull 6/11 5/11

#8:FAH is a 7-24-25 right triangle and

FH = AG (The diagonals in a rectangular are congruent) = AC = the radius of the circle = 25

So CF = 25 - 7 = 18 units

#10: Be careful of the unit measures. A lot of the times they are not the same so you need to do some conversions before getting the right answer.

18 inches = 18 /12 or 1.5 feet

[(24 x 32 x 1.5) - the volume of the water x 7.5 (gallons)] divided by (3 x 8 ) = 360 minutes

You can cancel "24" on both the numerator and denominator first.

Some Tircky Mathcounts School Questions: 2007 School

2011-01-28T07:23:00.001-05:00

2007 School Round Sprint:

#27. When the positive integers are arranged in order, filling in the successive diagonals of an infinite grid from top to bottom, as shown, the integer 41 is in the (5, 5) spot. What integer would we see in the (10, 10) spot if the rest of the grid were visible?

# 29. We know the following to be true:
1. X and K are integers with 500 < Z < 1000 and K > 1,
2. Z = K x K²
What is the value of K for which Z is a perfect square?

# 30: See this link.

# 6 Target: The Hubble constant, H0, is the ratio of the recessional velocity of a galaxy to its distance. The Hubble constant is estimated to be 70 km/sec per million parsecs of distance. (One parsec is equal to 3.26 light years.) Based on this ratio, how many billions of light years away would a galaxy be if it had a recessional velocity of 300,000 km/sec? Express your answer to the nearest whole number.

Solution:

#27: The easiest way to solve this question is to use the diagonal line: 1→5→13→25→41 (5,5) →61(6,6)→85→113→145→181(10,10)
Each progression is a multiple of 4. 4⁰ + 4¹ + 4² + ...4⁹ = 1 + 4 (1 + 2 + 3 +...9)= 1 + 180 = 181

#29: Z = K x K² = K ³ (given) and is also a perfect square.

Therefore, you need to find a number that is both a cube and a perfect square between 500 and 1000. Only 9³ = 729 works, so K = 9

# 6 Target: According to the given information. H = V/ D so D = V/H

D = (300,000 divided by 70) times (100000) times (3.26) divided by (1000, 000, 000) = 13.97142857 about 14 billions of light year

Some Tircky Mathcounts Chapter Questions: 2007 Chapter

2011-01-27T09:25:00.001-05:00

#26: Either increasing the radius or the height of a cylinder by six inches will result in the same volume. The original height of the cylinder is two inches. What is the original radius?

#27: Consider this pattern where the positive, odd integers are arranged in a triangular formation. The 1st through 4th rows are shown; each row has one more entry than the previous row. What is the sum of the integers in the 15th, row?
1

3 5

7 9 1

13 15 17 19

.

#28. Four couples are at a party. Four people of the eight are randomly selected to win a price. No person win more than one prize. What is the probability that both members of at least one couple win a prize? Express your answer as a common fraction.

#30. In parallelogram ABCD, AB = 16 cm, DA = 3√ 2 cm, and sides AB and DA form a 45-degree interior angle. In isosceles trapezoid WXYZ with WX ≠ YZ, segment WX is the longer parallel side and has length 16 cm, and two interior angles each have a measure of 45 degrees. Trapezoid WXYZ has the same area as parallelogram ABCD. What is the length of segment YZ?

Solutions:

#26: Let r be the original radius. (r + 6) ² x 2 = r² (2 + 6); (r² + 12r + 36) x 2 = 8r²

r² + 12r +36 = 4r² ; r² - 4r - 12 = 0 ; (r -6)(r+2) = 0; r = 6 or r = -2

#27: Solution 1:

The sum of the first "n" natural odd numbers = n²

There are (15 x 16)/2 = 120 odd numbers for the total 15 rows and (14 x 15)/2 = 105 odd numbers for the total 14 rows

120² - 105² = (120 + 105) (120 - 105) = 225 x 15 = 3375

Solution 2:

The sum of the first row = 1 or 1³

The sum of the second row = 8 or 2³

The sum of the third row = 27 or 3³

...

The sum of the 15th row = 15³ = 3375

#28: Use complementary counting: total ways minus no couple wins a prize.

(8 x 7 x 6 x 5 - 8 x 6 x 4 x 2) / 8 x 7 x 6 x 5 = 27/35

#30: The area of the parallelogram is 3√ 2 x 8√ 2 = 48

The area of the trapezoid =

1/2 [16 + (16 - 2h)] x h = 48

16h + 16h -2h² = 96; h² - 16h + 48 = 0; (h-4)(h-12) = 0; h = 4 or 12

Only h = 4 works so

YZ = 16 - 2 x 4 = 8

Geometry: Pythagorean Theory, Angle Ratio

2011-01-26T16:29:00.000-05:00

Using 45-45-90 degree angle ratios, which is 1-1-√ 2 , you can very easily solve this problems.

A lot of students thought this one is a 30-60-90 degree triangle. Well, it's not so you need to use the Pythagorean Theorem to solve for x or r according to what's given.

Some Tirckier Mathcounts Chapter Questions: 2007 Chapter on rate

2011-01-25T14:47:00.002-05:00

#24 2007 Chapter Sprint : Grady rides his bike 60% faster than his little brother Noah. If Grady rides 12 miles further than Noah in two hours, how fast does Noah ride?

#8 2007 Chapter Target: Dr. Lease leaves his house at exactly 7:20 a.m. every morning. When he averages 45 miles per hour, he arrives at his workplace five minutes late. When his averages 63 miles per hour, he arrives five minutes early. What speed should Dr. Lease average to arrive at his workplace precisely on time? Express your answer as a decimal to the nearest tenth.

#12 2002 AMC 10 A: Mr. Earl E. Bind leaves for work at exactly 8:00 A. M. every morning. When he average 40 miles per hour, he arrives at his workplace three minutes late. When he averages 60 miles per hour, he arrives three minutes early. At what average speed, in miles per hour, should Mr. Bird drive to arrive at his workplace precisely on time?

#10 2007 Chapter Team: Bricklayer Ben places 42 bricks per hour. Bricklayer Bob places 36 bricks per hour. Bricklayer Bob worked twice as many hours as Bricklayer Ben, and the two of them placed a total of 1254 bricks. How many bricks did Bricklayer Ben place?

Solutions:

#24: If Noah's bike speed is x m/h, every hour Grady will exceed his little brother Noah 3x/5 m/h (60% faster).

In two hours, Grady will exceed 3x/5 times 2 or 6x/5, which would equal 12 miles.

x = 12 x 5/6 = 10 m/h

#8: Solution I: Let t be the hour it took Dr. Lease to arrive at his workplace on time.

45(t + 1/12) = 63( t - 1/12) --- Make sure to convert 5 minutes to 1/12 hour since t represents hours.

Both sides times 12 and you have 45x12t + 45 = 63x12t - 63; 108 = 18x12t; t = 1/2

45 times (1/2 + 1/12) = 26.5 The distance of the commute.

26.5 divided by 1/2 hour = 52.5 m/h the right speed (D/T= Rate or Speed)

Solution II: Since r times t = d, r and d are directly proportional.

45: 63 = 5 : 7 (ratio of the rates) The same goes to time ratio 5: 7

63 - 45 = 18

45 + 5/12 times 18 = 52.5 m/h

or 63 - 7/12 times 18 = 52.5 m/h
This method works only when the time he arrives early and the time he arrives late are the same.

# 12 AMC 10-A -- Using the same methods as the previous question, the answer is 48 m/h

#10: Ben's rate is 42 b/h (brick per hour) and Bob 36 b/r. --given

Their time ratio is Ben to Bob = x : 2x

Since rate times time equal the output you have 42x + 36 times 2x = 1254

42x + 72x = 1254; x = 11 so Ben placed 11 x 42 = 462 bricks

AMC question on problem solving

2011-01-25T09:37:00.001-05:00

#25: 2006 AMC-10B
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right, "replies Mr. Jones, "and the last two digits just happens to be my age." Which of the following is not the age of one of Mr. Jones's children?
A. 4    B. 5    C. 6    D. 7    E. 8

#22: 2000 AMC-10 test
One morning each member of Angela's family drank an 8-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of total amount of mild and a sixth of the total amount of coffee. How many people are in the family?
A. 3            B. 4            C. 5            D. 6            E. 7

Solution:

Since the age of the oldest kid is 9, which means that the 4-digit number is divisible by 9.

Since each of two digits appears two times, the 4 digit could be 9900, 8811, 7722, 6633, 5544 or the previous 4-digit numbers arrangements.

Since 4, 6, 8 are all even numbers, it shows that the unit digit has to be even.

For a number to be divisible by 5, the unit digit could only be 0 or 5. Eliminate 5 and the only possible 4 digit arrangements to be divisible by 5 would be 9900 (Mr. Jones's age can't be 0.) or 9090, a number not divisible by 4 and 8. Thus 5 is the answer.

Further investigation will show that 5544 works for each of the other numbers.

#22: Let the whole family drank x cups of milk and y cups of coffee and let z denote the number of people in the family.

For Angela's portion, you would have x/4 + y/6 = 8 = x +y / z
3x + 2y = 96----equation 1
x + y = 8z ------equation 2 times 2
2x + 2y = 16z----------equation 3
equation 1 - equation 3 and you have x = 96 - 16z
Solve for the equation and you have x = 16, y = 24 and z = 5

Similar Triangles

2011-01-24T11:12:00.000-05:00

Solution: Triangle ABC is a 6-8-10 right triangle and is similar to AXB.

AB/AC = 6/10 = AX/AB = AX/6 so AX =3.6

AM - AX = XM = 5 - 3.6 = 1.4 cm

Similar Triangles: Team question

2011-01-23T09:31:00.000-05:00

9. In the figure below, quadrilateral CDEG is a square with CD = 3, and quadrilateral BEFH is a rectangle. If EB = 5, how many units is BH? Express your answer as a mixed number

Triangle BED is a 3-4-5 right triangle and is similar to triangle GEF.

BE : ED = GE : EF = 5 : 3 = 3 : FE

EF = 9/5 = BH The answer

Chapter : Similar triangle quesiton

2011-01-22T16:47:00.000-05:00

Solution: Triangle ABD is similar to triangle FGD. AB : FG = BD : GD = 10 : 3 (Given)

Since BD + GD = 10 so DG = 3/13 times 10 = 30/13

The area of the shaded region = 3 x 3 - 3 x (30/13) x 1/2 = 72/13

Some Tirckier Mathcounts Chapter Questions: 2003 Chapter

2011-01-21T08:36:00.000-05:00

Solutions are right after the questions.
However, please try the questions at least twice before reading the solutions.

#24: Two cylinder cans have the same volume. The height of one can is triple the height of the other. If the radius of the narrower can is 12 units, how many units are in the length of the radios of the wider can? Express your answer in simplest radical form.

#26. What is the greatest possible number of points of intersection for eight distinct lines in a plane?

#27. When its digits are reversed, a particular positive two-digit integer is increased by 20%. What is the original number?

#29. Each day, two out of the three teams in a class are randomly selected to participate in a MATHCOUNTS trial competition. What is the probability that Team A is selected on at least two
of the next three days? Express your answer as a common fraction.

#30. What is the sum of all of the four-digit positive integers that can be written with the digits 1, 2, 3 and 4 if each digit must be used exactly once in each four-digit positive integer?

#4: Target: A parallelogram has vertices at (0,0), (2, 5), (m, n) and (10, 0), where m and n are both positive numbers. How many units are in the length of the longest diagonal of the parallelogram?

#6: There is a 4 by 4 grid of points. Each point is one unit from its nearest horizontal and vertical neighbors. What is the greatest number of segments that can be drawn, using pairs of these points as endpoints, such that no two segments are the same length?

Solutions:

#24: This type of geometry questions are very common and simple. Just remind yourself not to solve the equation but simplify first as much as you can.

Make the height to the wider can "h" and the height to the narrow can "3h".

Make the radius of the wider can r and now you can set the two volume equal. (Given)

π r²h = 12²π x 3h; Cancel the π and the h and you have r² = 12²x 3; r = 12√ 13

#26: Two lines at most intersect at one point, three lines at most 3 points and four lines at most 6 points.

1,3, and 6 are triangle numbers and you can generate them using sum of the first "n" natural numbers.

8 lines would at most intersect at (1 + 2 + 3 ...7) = 28 points

#27: Set the two digit numbers 10x + y. According to the given information, you'll have

10y + x (reversed) = 6/5(10x + y) ; 50y + 5x = 60x + 6y; 44y = 55 x so 4y = 5x,

which means that y = 5 and x = 4

The original number is 45

#29: Hmmm...Two out of three teams go to the Mathcounts trial competition.

There are three teams A, B, C so 3C2 is 3, which means that Team A will be chosen 2/3 of the time.

The probability that Team A will be selected 2 of the 3 days is 2/3 times 2/3 times 1/3 (not selected) x 3 because it can be arranged as : A-A-NotA, A-NotA-A or NotA-A-A = 12/27

The probability that Team A will be selected 3 of the next three days is (2/3)³= 8/27

Sum them up and you get 20/27

#30: Solution I: You have 4! = 24 ways to rotate 1, 2, 3, 4 and since each number appears the same times, the average is 2.5.

2,5 x (1000 +100+10+1) x 24 = 2.5 x 1111 x 24 = 66660

Solution II: Each number will appear 6 times and (1+2+3+4) = 10

6 times 10 = 60, 60(1000 + 100 + 10 + 1) = 66660

#4: Target:

The answer is 13 units long.

#6: The longest distance you can draw horizontally or vertically is 3 units long.

So 1, 2, or 3 units long are some possibility.

Now looks at the possible diagonal length. Do it systematically. The possible lengths you can have are:

√ 2 (1, 1 as the other two legs)

√ 8 (2, 2 as the other two legs)

√ 18 (3, 3 as the other two legs)

√ 5 (1, 2 as the other two legs)

√ 10 (1, 3 as the other two legs)

√ 13 (2, 3 as the other two legs) All together you have 9 possible segments.

Some Tirckier Mathcounts Chapter Questions: 2005 Chapter

2011-01-19T22:42:00.003-05:00

Here are some tricky questions on 2005 Mathcounts Chapter Sprint.
Solutions are provided below but please refrain yourself from reading them too soon.
Try at least twice before looking for faster/easier ways to tackle these problems.
Hope this helps. Happy problem solving!!

#17: What is the sum of all the district positive two-digit factors of 144?

#18: The points B(1, 1), I(2, 4) and G(5, 1) are plotted in the standard rectangular coordinate system to form triangle BIG. Triangle BIG is translated five units to the left and two units upward to triangle BTG', in such a way that B' is the image of B, I' is the image of I, and G' is the image of G. What is the midpoint of segment B'G'? Express your answer as an ordered pair.

#22: How many ordered pairs (x, y) satisfy BOTH conditions below? Condition I: x = 1 or y = 0 or y = 2
Condition II: x = 0 or x = 2 or y = 1

#27: The function f is defined by f (n) = f (n − 1) + f (n − 2). It is also true that f (1) = 3 and f (3) = 10. What is the value of f (6)?

#28. If a/b = 3/4; b/c = 8/9 and c/d = 2/3,, what is the value of ad/ b²? Express your answer as a common fraction.

#29. A play has two male roles, two female roles and two roles that can be either gender. Only a man can be assigned to a male role, and only a woman can be assigned to a female role. If five men and six women audition, in how many ways can the six roles be assigned?

#30. What is the arithmetic mean of all of the positive two-digit integers with the property that the integer is equal to the sum of its first digit plus its second digit plus the product of its two digits?

#5: Target: What is the greatest whole number that must be a factor of the sum of any four consecutive positive odd numbers?

#17: Solution:

144 = 2⁴ x 3²

Make a list (See image on the left.) and add up all the two digit numbers.

The answer is 226.

#18: Solution: This one looks complicated but actually is very easy if you first find the midpoint of BG before shifting.

To find the midpoint, add up the two end points and divided by 2 so the midpoint of BG is (3, 1).

After translating 5 units to the left and 2 units upward, you got B'G' = (-2, 3)

#22: For condition I, you can have (1, ), ( , 0) or ( , 2).
For condition II, you can have (0, ), (2, ) or ( , 1 ).
Combine the possible ways to satisfy both conditions and you have 5 pairs, which are (1, 1), (0, 0), (0, 2),
(2, 0) and (2, 2)

#27: Solution: This kind of question requires you to use the given information to deduct the new value, so keep on generating new values till the desired outcome.
f(1) = 3 and f(3) = 10 (given) From this and the rule you get f(3) = f(2) + f(1); 10 = 3 + f(2) so f(2) = 7
f(4) = f(3) + f(2) = 10 + 7 = 17; f(5) = f(4) + f(3) = 17 + 10 = 27; f(6) = f(5) + f(4) = 27 + 17 = 44

#28: Don't do busy work. Observe the wanted equation and see if you can use the given to achieve that outcome?
a/b times d/c times c/b gives you exactly that so 3/4 times 3/2 (flip the numbers since you want d/c) times 9/8
(again, flip the numerator and denominator) = 81/64

#29: This one is tricky in that each role is different so you need to use permutation.

There are four cases for the last part; two roles could be played by either gender.
They are :
case 1: both were played by men
case 2: both were played by women
case 3: first role was played by a man and second was played by a woman
case 4: first role was played by a woman and second was played by a man

Ok, now we can add them up;case 1: 5 x 4 (men) x 6 x 5 (women) x 3 x 2 = 3600
case 2: 5 x 4 x 6 x 5 x 4 x 3 = 7200
case 3: 5 x 4 x 6 x 5 x 3 x 4 = 7200
case 4: 5 x 4 x 6 x 5 x 4 x 3 = 7200
The sum is 25,200

#30: A very important technique is to set up a two-digit number. You use 10a + b.
According to the given, you have 10a + b = a + b + ab
Simplify and you get 9a - ab = a (9-b) = 0, which means that either a = 0 (you can't have tenth digit equal "0")
or b - 9 = 0 so b = 9.
The two-digit numbers all have the unit digit 9 so they are 19, 29, ...99.
The arithmetic mean is the median since there are 9 numbers so the answer is 59.

#5: Credit: Solutions are from this newsletter.

Proof: A general expression for the sum (S) of any four consecutive positive odd numbers may be written as,S=(2n+1) + (2n+3) + (2n+5) + (2n+7) = 8n + 16 = 8(n+2)
where n is an integer greater than or equal to 0. Thus, 8 is a factor of the sum of any four consecutive positive odd numbers.
It remains to be proven that 8 is the greatest whole number that is a factor. To show this, consider the case when S takes on
the smallest possible value. That occurs when n=0 and S=1+3+5+7 =16. The only factors of 16 are 1, 2, 4, 8, and 16. The factors
1, 2, and 4 may be disregarded because they are less than 8. This leaves only 16 as a possibility for the greatest whole
number factor. Now consider the case when n=1 and S =3+5+7+9=24. The number 16 is not a factor of 24. Thus, 8 is the greatest
whole number that must be a factor of the sum of any four consecutive positive odd numbers.
We received solutions to this puzzler from Glenn Ballard (MA, 1967), Clark C. Bond (BA, 1960), and Mary Krimmel (MA, 1970).