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. Unknown error_msg)
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 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |