Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

(→‎Problem)
(→‎Solution)
Line 17: Line 17:
  
 
==Solution==
 
==Solution==
−
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>.  
+
Let <math>AB = \Gamma \Delta = x</math>, <math>B\Gamma = A\Delta = y</math>. Using the [[Pythagorean Theorem]], <math>KM = \sqrt{\frac{x^2}{9} + y^2}</math>, <math>\Lambda N = \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>.  
  
−
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.  
+
Be careful not to just try a couple of simple examples like <math>AB\Gamma \Delta</math> being a square, where we will get the answer <math>5/9</math>, which is incorrect in general.
  
 
==See also==
 
==See also==

Revision as of 21:04, 17 October 2007

Problem

2006 CyMO-22.PNG

$AB\Gamma \Delta$ is rectangular and the points $K,\Lambda ,M,N$ lie on the sides $AB, B\Gamma, \Gamma \Delta , \Delta A$ respectively so that $\frac{AK}{KB}=\frac{B\Lambda}{\Lambda \Gamma}=\frac{\Gamma M}{M\Delta}=\frac{\Delta N}{NA}=2$. If $E_1$ is the area of $K\Lambda MN$ and $E_2$ is the area of the rectangle $AB\Gamma \Delta$, 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 $AB = \Gamma \Delta = x$, $B\Gamma = A\Delta = y$. Using the Pythagorean Theorem, $KM = \sqrt{\frac{x^2}{9} + y^2}$, $\Lambda N = \sqrt{x^2 + \frac{y^2}{9}}$. 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 $AB\Gamma \Delta$ 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
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2006_Cyprus_MO/Lyceum/Problem_22&oldid=18295"