2017 AMC 10A Problems/Problem 23
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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 triangles. However, some of them are degenerate if
or more points lie on the same line. There are a total of 12 lines that go through 5 points (5 vertical, 5 horizontal, 2 diagonal), 4 lines with slope
that go through exactly 4 points, and 16 lines (
) with slope
. Subtracting these degenerate triangles, we get an answer of
.