Difference between revisions of "2009 AIME I Problems/Problem 11"
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== Solution == | == Solution == | ||
+ | Solution 1 (This solution requires linear algeber knowledgw) | ||
+ | |||
+ | Let the two points be point P and Q | ||
+ | |||
+ | and <math>P=(x_1,y_1),Q=(x_2,y_2) | ||
+ | |||
+ | We cna 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 | ||
+ | |||
+ | <cmath>(x_1)(y_2)-(x_2)(y_1)=(x_1)(2009-41(x_2))-(x_2)(2009-41(x_1))</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, </math>((x_1)-(x_2))<math> must be even | ||
+ | |||
+ | Thus the two x's have to be both odd or even. | ||
+ | |||
+ | Also note that the maximum value for x is </math>49<math> and minimum is </math>0<math>. | ||
+ | |||
+ | There are </math>25<math> even and </math>25<math> odd number | ||
+ | |||
+ | Thus, there are | ||
+ | |||
+ | </math>(_{25}C_2)+(_{25}C_2)=\boxed{600}$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 22:27, 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 $P=(x_1,y_1),Q=(x_2,y_2)
We cna calculate the area of the parallelogram span with the determinant of matrix PQ, P above Q, since triangle is half of the area of the parallelogram. We just need the determinant to be even
The deteminant is
<cmath>(x_1)(y_2)-(x_2)(y_1)=(x_1)(2009-41(x_2))-(x_2)(2009-41(x_1))</cmath>
<cmath>=2009(x_1)-41(x_1)(x_2)-2009(x_2)+41(x_1)(x_2)=2009((x_1)-(x_2))</cmath>
since 2009 is not even,$ (Error compiling LaTeX. ! Missing $ inserted.)((x_1)-(x_2))$must be even
Thus the two x's have to be both odd or even.
Also note that the maximum value for x is$ (Error compiling LaTeX. ! Missing $ inserted.)490$.
There are$ (Error compiling LaTeX. ! Missing $ inserted.)2525$odd number
Thus, there are$ (Error compiling LaTeX. ! Missing $ inserted.)(_{25}C_2)+(_{25}C_2)=\boxed{600}$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 |