# Difference between revisions of "2009 AIME I Problems/Problem 11"

## Problem

Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $41x + y = 2009$. 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. ! Missing \$ inserted.)((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. ! Missing \$ inserted.)49 $and minimum is$0\$.

There are\$ (Error compiling LaTeX. ! Missing \$ inserted.)25 $even and$25\$odd number

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