HMMT 二月 2015 · 冲刺赛 · 第 6 题

HMMT February 2015 — Guts Round — Problem 6

[ 5 ] Let D be a regular ten-sided polygon with edges of length 1. A triangle T is defined by choosing three vertices of D and connecting them with edges. How many different (non-congruent) triangles T can be formed?

解析

[ 5 ] Let D be a regular ten-sided polygon with edges of length 1. A triangle T is defined by choosing three vertices of D and connecting them with edges. How many different (non-congruent) triangles T can be formed? Answer: 8 The problem is equivalent to finding the number of ways to partition 10 into a sum of three (unordered) positive integers. These can be computed by hand to be (1 , 1 , 8), (1 , 2 , 7), (1 , 3 , 6), (1 , 4 , 5), (2 , 2 , 6), (2 , 3 , 5), (2 , 4 , 4), (3 , 3 , 4).