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2009 AIME I Problems/Problem 4: Difference between revisions

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One of the ways to solve this problem is to make this parallelogram a straight line.
One of the ways to solve this problem is to make this parallelogram a straight line.


So the whole length of the line<math>APC, or AMC</math> is <math>1000x+2009x=3009x</math>
So the whole length of the line<math>APC(AMC or ANC), and ABC</math> is <math>1000x+2009x=3009x</math>


And <math>AP</math> will be <math>17x</math>
And <math>AP(AM or AN)</math> is <math>17x</math>


So the answer is <math>3009/17 = 177</math>
So the answer is <math>3009/17 = 177</math>

Revision as of 19:20, 20 March 2009

Problem 4

In parallelogram $ABCD$, point $M$ is on $\overline{AB}$ so that $\frac {AM}{AB} = \frac {17}{1000}$ and point $N$ is on $\overline{AD}$ so that $\frac {AN}{AD} = \frac {17}{2009}$. Let $P$ be the point of intersection of $\overline{AC}$ and $\overline{MN}$. Find $\frac {AC}{AP}$.

Solution

Solution

One of the ways to solve this problem is to make this parallelogram a straight line.

So the whole length of the line$APC(AMC or ANC), and ABC$ is $1000x+2009x=3009x$

And $AP(AM or AN)$ is $17x$

So the answer is $3009/17 = 177$

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