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Difference between revisions of "2006 Cyprus MO/Lyceum/Problem 22"

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==Problem==
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== Problem ==
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[[Image:2006 CyMO-22.PNG|250px]]
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<math>ABCD</math> is rectangular and the points <math>K,L,M,N</math> lie on the sides <math>AB, BC, CD, DA</math> respectively so that <math>\frac{AK}{KB}=\frac{BL}{LC}=\frac{CM}{MD}=\frac{DN}{NA}=2</math>. If <math>E_1</math> is the area of <math>KLMN</math> and <math>E_2</math> is the area of the rectangle <math>ABCD</math>, the ratio <math>\frac{E_1}{E_2}</math> equals
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A. <math>\frac{5}{9}</math>
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B. <math>\frac{1}{3}</math>
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C. <math>\frac{9}{5}</math>
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D. <math>\frac{3}{5}</math>
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E. None of these
  
 
==Solution==
 
==Solution==
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{{solution}}
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Let <math>AB = CD = x</math>, <math>BC = AD = y</math>. Using the [[Pythagorean Theorem]], <math>KM = \sqrt{\frac{x^2}{9} + y^2}</math>, <math>LN = \sqrt{x^2 + \frac{y^2}{9}}</math>. Using the formula <math>A = \frac{1}{2}d_1d_2</math> for a [[rhombus]], we get <math>\frac{1}{2}\sqrt{\left(x^2 + \frac{y^2}{9}\right)\left(x^2 + \frac{y^2}{9}\right)} = \frac{1}{2}\sqrt{\frac{x^4}{9} + \frac{y^4}{9} + \frac{82}{81}x^2y^2} = \frac{\sqrt{9x^4 + 9y^4 + 82x^2y^2}}{18}</math>. Thus the ratio is <math>\frac{\sqrt{9x^4 + 9y^4 + 82x^2y^2}}{18xy}</math>. There is no way we can simplify this further, and in fact we can plug in different values of <math>x,y</math> to see that the answer is <math>\mathrm{E}</math>.
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Be careful not to just try a couple of simple examples like <math>ABCD</math> being a square, where we will get the answer <math>5/9</math>, which is incorrect in general.
  
 
==See also==
 
==See also==
 
{{CYMO box|year=2006|l=Lyceum|num-b=21|num-a=23}}
 
{{CYMO box|year=2006|l=Lyceum|num-b=21|num-a=23}}
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[[Category:Introductory Geometry Problems]]

Revision as of 18:09, 15 October 2007

Problem

2006 CyMO-22.PNG

$ABCD$ is rectangular and the points $K,L,M,N$ lie on the sides $AB, BC, CD, DA$ respectively so that $\frac{AK}{KB}=\frac{BL}{LC}=\frac{CM}{MD}=\frac{DN}{NA}=2$. If $E_1$ is the area of $KLMN$ and $E_2$ is the area of the rectangle $ABCD$, the ratio $\frac{E_1}{E_2}$ equals

A. $\frac{5}{9}$

B. $\frac{1}{3}$

C. $\frac{9}{5}$

D. $\frac{3}{5}$

E. None of these

Solution

Let $AB = CD = x$, $BC = AD = y$. Using the Pythagorean Theorem, $KM = \sqrt{\frac{x^2}{9} + y^2}$, $LN = \sqrt{x^2 + \frac{y^2}{9}}$. Using the formula $A = \frac{1}{2}d_1d_2$ for a rhombus, we get $\frac{1}{2}\sqrt{\left(x^2 + \frac{y^2}{9}\right)\left(x^2 + \frac{y^2}{9}\right)} = \frac{1}{2}\sqrt{\frac{x^4}{9} + \frac{y^4}{9} + \frac{82}{81}x^2y^2} = \frac{\sqrt{9x^4 + 9y^4 + 82x^2y^2}}{18}$. Thus the ratio is $\frac{\sqrt{9x^4 + 9y^4 + 82x^2y^2}}{18xy}$. There is no way we can simplify this further, and in fact we can plug in different values of $x,y$ to see that the answer is $\mathrm{E}$.

Be careful not to just try a couple of simple examples like $ABCD$ being a square, where we will get the answer $5/9$, which is incorrect in general.

See also

2006 Cyprus MO, Lyceum (Problems)
Preceded by
Problem 21 		Followed by
Problem 23
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
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