Difference between revisions of "2009 AIME I Problems/Problem 11"
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and <math>P=(x_1,y_1),Q=(x_2,y_2)</math> | and <math>P=(x_1,y_1),Q=(x_2,y_2)</math> | ||
− | We can calculate the area of the parallelogram span with the determinant of matrix | + | We can calculate the area of the parallelogram span with the determinant of matrix |
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+ | <math>\det ({\matrix {P \above Q}})=\det </math><math>({\matrix {x_1 \above x_2} \right \matrix {y_1 \above y_2})</math> | ||
+ | |||
+ | since triangle is half of the area of the parallelogram. We just need the determinant to be even | ||
The deteminant is | The deteminant is |
Revision as of 22:39, 20 March 2009
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
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
since 2009 is not even, must be even
Thus the two x's have to be both odd or even.
Also note that the maximum value for x is and minimum is .
There are even and odd number
Thus, there are
of such triangle
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 |