2009 AIME I Problems/Problem 11
Problem
Consider the set of all triangles where
is the origin and
and
are distinct points in the plane with nonnegative integer coordinates
such that
. Find the number of such distinct triangles whose area is a positive integer.
Solution
Let the two points and
be defined with coordinates;
and
We can calculate the area of the parallelogram with the determinant of the matrix of the coordinates of the two points(shoelace theorem).
$\det \left({\matrix {P \above Q}}\right)=\det \left({\matrix {x_1 \above x_2}\matrix {y_1 \above y_2}\right).$ (Error compiling LaTeX. Unknown error_msg)
Since the triangle has half the area of the parallelogram, we just need the determinant to be even.
The determinant is
Since is not even,
must be even, thus the two
's must be of the same parity. Also note that the maximum value for
is
and the minimum is
. There are then
even and
odd numbers available for use as coordinates and thus there are
such triangles.
See also
2009 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |