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

Solution 1 (This solution requires linear algeber knowledgw)

Let the two points be point P and Q

and $P=(x_1,y_1),Q=(x_2,y_2)

We cna calculate the area of the parallelogram span with the determinant of matrix PQ, P above Q, since triangle is half of the area of the parallelogram. We just need the determinant to be even

The deteminant is

<cmath>(x_1)(y_2)-(x_2)(y_1)=(x_1)(2009-41(x_2))-(x_2)(2009-41(x_1))</cmath>

<cmath>=2009(x_1)-41(x_1)(x_2)-2009(x_2)+41(x_1)(x_2)=2009((x_1)-(x_2))</cmath>

since 2009 is not even,$ (Error compiling LaTeX. Unknown error_msg)((x_1)-(x_2))$must be even

Thus the two x's have to be both odd or even.

Also note that the maximum value for x is$ (Error compiling LaTeX. Unknown error_msg)490$.

There are$ (Error compiling LaTeX. Unknown error_msg)2525$odd number

Thus, there are$ (Error compiling LaTeX. Unknown error_msg)(_{25}C_2)+(_{25}C_2)=\boxed{600}$of such triangle