HMMT 二月 2012 · TEAM2 赛 · 第 2 题

HMMT February 2012 — TEAM2 Round — Problem 2

[ 10 ] You are given two line segments of length 2 for each integer 0 ≤ n ≤ 10. How many distinct nondegenerate triangles can you form with three of the segments? Two triangles are considered distinct if they aren’t congruent.

解析

[ 10 ] You are given two line segments of length 2 for each integer 0 ≤ n ≤ 10. How many distinct nondegenerate triangles can you form with three of the segments? Two triangles are considered distinct if they are not congruent. a b c Answer: 55 First, observe that if we have three sticks of distinct lengths 2 < 2 < 2 , then a b b +1 c 2 + 2 < 2 ≤ 2 , so we cannot form a triangle. Thus, we must have (exactly) two of our sticks the a a b same length, so that our triangle has side lengths 2 , 2 , 2 . This triangle is non-degenerate if and only ( ) 11 a +1 b if 2 > 2 , and since a 6 = b , this happens if and only if a > b . Clearly, there are = 55 ways to 2 choose such a, b .