Track / Stage 4 / 233 of 340 #493 of 1964

Problem 493

AMC 12 late, AIME early

Geometry Difficulty 4.9 Find the answer HMMT_2

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?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Official solution

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 $(1,1,8),(1,2,7),(1,3,6),(1,4,5),(2,2,6),(2,3,5),(2,4,4),(3,3,4)$

Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.