2010 Mathcounts School Sprint

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.