# 2017 AIME I Problems/Problem 1

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## Problem 1

Fifteen distinct points are designated on : the 3 vertices , , and ; other points on side ; other points on side ; and other points on side . Find the number of triangles with positive area whose vertices are among these points.

## Solution

Every triangle is uniquely determined by 3 points. There are ways to choose 3 points, but that counts the degenerate triangles formed by choosing three points on a line. There are invalid cases on segment , invalid cases on segment , and invalid cases on segment for a total of invalid cases. The answer is thus .