2009 AIME I Problems/Problem 11

Revision as of 22:41, 20 March 2009 by Moplam (talk | contribs) (Solution)

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 alg. knowledge)

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 (ProblemsAnswer KeyResources)
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
Invalid username
Login to AoPS