1993 AHSME Problems/Problem 28
Problem
How many triangles with positive area are there whose vertices are points in the -plane whose coordinates are integers
satisfying
and
?
Solution
The vertices of the triangles are limited to a grid, with
points total. Every triangle is determined by
points chosen from these
for a total of
. However, triangles formed by collinear points do not have positive area. For each column or row, there are
such degenerate triangles. There are
total columns and rows, contributing
invalid triangles. There are also
for both of the diagonals and
for each of the
shorter diagonals. There are a total of
invalid triangles counted in the
, so the answer is
or answer choice
.
See also
1993 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 27 |
Followed by Problem 29 | |
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