2017 AMC 10A Problems/Problem 23
Problem
How many triangles with positive area have all their vertices at points in the coordinate plane, where
and
are integers between
and
, inclusive?
Solution
There are a total of sets of three points. However, some of them form degenerate triangles (i.e., they have area of 0) if the three points are collinear. There are a total of 12 lines that go through 5 points (5 vertical, 5 horizontal, 2 diagonal), which contributes
degenerate triangles, 4 linesthat go through exactly 4 points, which contributes
degenerate triangles, and 16 lines that go through exactly three points, which contributes
degenerate triangles. Subtracting these degenerate triangles, we get an answer of
.