2009 AIME I Problems/Problem 11
Contents
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 1
Let the two points and be defined with coordinates; and
We can calculate the area of the parallelogram with the determinant of the matrix of the coordinates of the two points(shoelace theorem).
$\det \left({\matrix {P \above Q}}\right)=\det \left({\matrix {x_1 \above x_2}\matrix {y_1 \above y_2}\right).$ (Error compiling LaTeX. ! Package amsmath Error: Old form `\matrix' should be \begin{matrix}.)
Since the triangle has half the area of the parallelogram, we just need the determinant to be even.
The determinant is
Since is not even, must be even, thus the two 's must be of the same parity. Also note that the maximum value for is and the minimum is . There are even and odd numbers available for use as coordinates and thus there are such triangles.
Solution 2
As in the solution above, let the two points and be defined with coordinates; and .
If the coordinates of and have nonnegative integer coordinates, and must be lattice points either
- on the nonnegative x-axis
- on the nonnegative y-axis
- in the first quadrant
We can calculate the y-intercept of the line to be and the x-intercept to be .
Using the point-to-line distance formula, we can calculate the height of from vertex (the origin) to be:
Let be the base of the triangle that is part of the line .
The area is calculated as:
Let the numerical area of the triangle be .
So,
We know that is an integer. So, , where is also an integer.
We defined the points and as and .
Changing the y-coordinates to be in terms of x, we get:
and .
The distance between them equals .
Using the distance formula, we get
WLOG, we can assume that .
Taking the last two equalities from the string of equalities and putting in our assumption that , we get
.
Dividing both sides by , we get
As we mentioned, is an integer, so is an even integer. Also, and are both positive integers. So, and are between 0 and 49, inclusive. Remember, as well.
- There are 48 ordered pairs such that their positive difference is 2.
- There are 46 ordered pairs such that their positive difference is 4.
...
- Finally, there are 2 ordered pairs such that their positive difference is 48.
Summing them up, we get that there are triangles.
See also
2009 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
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