Difference between revisions of "2009 AIME I Problems/Problem 11"
(→Solution) |
(→Solution) |
||
| Line 4: | Line 4: | ||
== Solution == | == Solution == | ||
| − | Let the two points | + | Let the two points <math>P</math> and <math>Q</math> be defined with coordinates; <math>P=(x_1,y_1)</math> and <math>Q=(x_2,y_2)</math> |
We can calculate the area of the parallelogram with the determinant of the matrix of the coordinates of the two points(shoelace theorem). | We can calculate the area of the parallelogram with the determinant of the matrix of the coordinates of the two points(shoelace theorem). | ||
Revision as of 17:47, 22 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
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
![\[(x_1)(y_2)-(x_2)(y_1)=(x_1)(2009-41(x_2))-(x_2)(2009-41(x_1))=2009(x_1)-41(x_1)(x_2)-2009(x_2)+41(x_1)(x_2)=2009((x_1)-(x_2))\]](http://latex.artofproblemsolving.com/e/3/3/e33298a2be6184ff8bbad54ccf1f10df5e7997df.png)
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 then 
even and odd numbers and thus there are 
such 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 | ||
