# 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 can calculate the area of the parallelogram span with the determinant of matrix

$\det ({\matrix {P \above Q}})=\det$ (Error compiling LaTeX. ! Package amsmath Error: Old form \matrix' should be \begin{matrix}.)$({\matrix {x_1 \above x_2} \right \matrix {y_1 \above y_2})$ (Error compiling LaTeX. ! Package amsmath Error: Old form \matrix' should be \begin{matrix}.)

since triangle is half of the area of the parallelogram. We just need the determinant to be even

The deteminant is

$$(x_1)(y_2)-(x_2)(y_1)=(x_1)(2009-41(x_2))-(x_2)(2009-41(x_1))$$

$$=2009(x_1)-41(x_1)(x_2)-2009(x_2)+41(x_1)(x_2)=2009((x_1)-(x_2))$$

since 2009 is not even, $((x_1)-(x_2))$ must be even

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

Also note that the maximum value for x is $49$ and minimum is $0$.

There are $25$ even and $25$ odd number

Thus, there are

$(_{25}C_2)+(_{25}C_2)=\boxed{600}$of such triangle

## See also

 2009 AIME I (Problems • Answer Key • Resources) Preceded byProblem 10 Followed byProblem 12 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 All AIME Problems and Solutions
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