Difference between revisions of "2009 AIME I Problems/Problem 11"

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== Solution ==
 
== Solution ==
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Solution 1 (This solution requires linear algeber knowledgw)
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Let the two points be point P and Q
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and <math>P=(x_1,y_1),Q=(x_2,y_2)
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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
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 +
The deteminant is
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<cmath>(x_1)(y_2)-(x_2)(y_1)=(x_1)(2009-41(x_2))-(x_2)(2009-41(x_1))</cmath>
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<cmath>=2009(x_1)-41(x_1)(x_2)-2009(x_2)+41(x_1)(x_2)=2009((x_1)-(x_2))</cmath>
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since 2009 is not even, </math>((x_1)-(x_2))<math> must be even
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Thus the two x's have to be both odd or even.
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Also note that the maximum value for x is </math>49<math> and minimum is </math>0<math>.
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There are </math>25<math> even and </math>25<math> odd number
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Thus, there are
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</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 $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $41x + y = 2009$. 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.)49$and minimum is$0$.

There are$ (Error compiling LaTeX. ! Missing $ inserted.)25$even and$25$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 (ProblemsAnswer KeyResources)
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
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