Notes on 2011 Mathcounts School Team Round Questions

#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.