Difference between revisions of "2009 AIME I Problems/Problem 11"
(→Solution) |
(→Solution) |
||
Line 8: | Line 8: | ||
Let the two points be point P and Q | Let the two points be point P and Q | ||
− | and <math>P=(x_1,y_1),Q=(x_2,y_2) | + | and <math>P=(x_1,y_1),Q=(x_2,y_2)</math> |
− | We | + | We can calculate the area of the parallelogram span with the determinant of matrix PQ, P above Q, since triangle is half of the area of the parallelogram. We just need the determinant to be even |
The deteminant is | The deteminant is | ||
Line 18: | Line 18: | ||
<cmath>=2009(x_1)-41(x_1)(x_2)-2009(x_2)+41(x_1)(x_2)=2009((x_1)-(x_2))</cmath> | <cmath>=2009(x_1)-41(x_1)(x_2)-2009(x_2)+41(x_1)(x_2)=2009((x_1)-(x_2))</cmath> | ||
− | since 2009 is not even, < | + | since 2009 is not even, <math>((x_1)-(x_2))</math> must be even |
Thus the two x's have to be both odd or even. | Thus the two x's have to be both odd or even. | ||
− | Also note that the maximum value for x is < | + | Also note that the maximum value for x is <math>49</math> and minimum is <math>0</math>. |
− | There are < | + | There are <math>25</math> even and <math>25</math> odd number |
Thus, there are | Thus, there are | ||
− | < | + | <math>(_{25}C_2)+(_{25}C_2)=\boxed{600}</math>of such triangle |
== See also == | == See also == | ||
{{AIME box|year=2009|n=I|num-b=10|num-a=12}} | {{AIME box|year=2009|n=I|num-b=10|num-a=12}} |
Revision as of 22:28, 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 PQ, P above Q, 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 |