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 
.
