(American Mathematics Contest 8) Solutions Pamphlet

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$File day 5 notes unit 2 math 8.pdf$

<strong>Solutions</strong> AMC 8 2005 4 12. (D) You can solve this problem by guessing and checking. If Big Al had eaten 10 bananas on May 1, then he would have eaten 10 + 16 + 22 + 28 + 34 = 110 bananas. This is 10 bananas too many, so he actually ate 2 fewer bananas each day. Thus, Big Al ate 8 bananas on May 1 and 32 bananas on May 5. OR The average number of bananas eaten per day was 100 5 = 20. Because the number of bananas eaten on consecutive days differs by 6 and there are an odd number of days, the average is also the median. Therefore, the average number of bananas he ate per day, 20, is equal to the number of bananas he ate on May 3. So on May 5 Big Al ate 20 + 12 = 32 bananas. OR Let x be the number of bananas that Big Al ate on May 5. The following chart documents his banana intake for the five days. May 5 May 4 May 3 May 2 May 1 x x − 6 x − 12 x − 18 x − 24 The total number of bananas Big Al ate was 5x − 60, which must be 100. So Big Al ate x = 160 5 = 32 bananas on May 5. 13. (C) ¡¢ © £¤¥¦§¨ Rectangle ABCG has area 8×9 = 72, so rectangle F EDG has area 72−52 = 20. The length of F G equals DE = 9 − 5 = 4, so the length of EF is 20 4 = 5. Therefore, DE + EF = 4 + 5 = 9. 14. (B) Each team plays 10 games in its own division and 6 games against teams in the other division. So each of the 12 teams plays 16 conference games. Because = 96 games scheduled. each game involves two teams, there are 12×16 2 15. (C) Because the perimeter of such a triangle is 23, and the sum of the two equal side lengths is even, the length of the base is odd. Also, the length of the base is less than the sum of the other two side lengths, so it is less than half of 23. Thus the six possible triangles have side lengths 1, 11, 11; 3, 10, 10; 5, 9, 9; 7, 8, 8; 9, 7, 7 and 11, 6, 6.
<strong>Solutions</strong> AMC 8 2005 5 16. (D) It is possible for the Martian to pull out at most 4 red, 4 white and 4 blue socks without having a matched set. The next sock it pulls out must be red, white or blue, which gives a matched set. So the Martian must select 4 × 3 + 1 = 13 socks to be guaranteed a matched set of five socks. 17. (E) Evelyn covered more distance in less time than Briana, Debra and Angela, so her average speed is greater than any of their average speeds. Evelyn went almost as far as Carla in less than half the time that it took Carla, so Evelyn’s average speed is also greater than Carla’s. OR The ratio of distance to time, or average speed, is indicated by the slope of the line from the origin to each runner’s point in the graph. Therefore, the line from the origin with the greatest slope will correspond to the runner with the greatest average speed. Because Evelyn’s line has the greatest slope, she has the greatest average speed. 18. (C) The smallest three-digit number divisible by 13 is 13×8 = 104, so there are seven two-digit multiples of 13. The greatest three-digit number divisible by 13 is 13 × 76 = 988. Therefore, there are 76 − 7 = 69 three-digit numbers divisible by 13. OR Because the integer part of 999 13 is 76, there are 76 multiples of 13 less than or equal to 999. Because the integer part of 99 13 is 7, there are 7 multiples of 13 less than or equal to 99. That means there are 76 − 7 = 69 multiples of 13 between 100 and 999. 19. (A) ¡ By the Pythagorean Theorem, AE = √ 302 − 242 = √ 324 = 18. (Or note that triangle AEB is similar to a 3-4-5 right triangle, so AE = 3 × 6 = 18.) Also CF = 24 and F D = √ 25 2 − 24 2 = √ 49 = 7. The perimeter of the trapezoid is 50 + 30 + 18 + 50 + 7 + 25 = 180. ¥¦§ ¢¡£¤£ ¨©£¤
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