Showing posts with label 2011 Mathcounts Chapter Sprint solutions. Show all posts
Showing posts with label 2011 Mathcounts Chapter Sprint solutions. Show all posts

Saturday, September 1, 2018

2011 Mathcounts Chapter Sprint Round solutions

#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) * 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 .