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 "2007 Cyprus MO/Lyceum/Problem 11"

(→‎Solution: You cannot use calculators in this competition (This was not a nescessary operation))
(→‎Solution)
Line 7: Line 7:
 
<math>X=\frac{1}{2007 \sqrt{2006}+2006 \sqrt{2007}}</math>
 
<math>X=\frac{1}{2007 \sqrt{2006}+2006 \sqrt{2007}}</math>
 
<math>=\frac{1}{2007 \sqrt{2006}+2006 \sqrt{2007}}\cdot\frac{2007 \sqrt{2006}-2006 \sqrt{2007}}{2007 \sqrt{2006}-2006 \sqrt{2007}}</math>
 
<math>=\frac{1}{2007 \sqrt{2006}+2006 \sqrt{2007}}\cdot\frac{2007 \sqrt{2006}-2006 \sqrt{2007}}{2007 \sqrt{2006}-2006 \sqrt{2007}}</math>
−
<math>=\frac{2007\sqrt{2006}-2006\sqrt{2007}}{2006 \times 2007}</math>
+
<math>=\frac{2007\sqrt{2006}-2006\sqrt{2007}}{2006 \cdot 2007}</math>
−
<math>Y=\frac{1}{\sqrt{2006}}-\frac{1}{\sqrt{2007}}=\frac{\sqrt{2006}}{2006}-\frac{\sqrt{2007}}{2007}=\frac{2007\sqrt{2006}-2006\sqrt{2007}}{2006 \times 2007}</math>
+
<math>Y=\frac{1}{\sqrt{2006}}-\frac{1}{\sqrt{2007}}=\frac{\sqrt{2006}}{2006}-\frac{\sqrt{2007}}{2007}=\frac{2007\sqrt{2006}-2006\sqrt{2007}}{2006 \cdot 2007}</math>
  
 
<math>X=Y\Rightarrow\mathrm{ C}</math>
 
<math>X=Y\Rightarrow\mathrm{ C}</math>

Revision as of 13:29, 6 May 2007

Problem

If $X=\frac{1}{2007 \sqrt{2006}+2006 \sqrt{2007}}$ and $Y=\frac{1}{\sqrt{2006}}-\frac{1}{\sqrt{2007}}$, which of the following is correct?

$\mathrm{(A) \ } X=2Y\qquad \mathrm{(B) \ } Y=2X\qquad \mathrm{(C) \ } X=Y\qquad \mathrm{(D) \ } X=Y^2\qquad \mathrm{(E) \ } Y=X^2$

Solution

$X=\frac{1}{2007 \sqrt{2006}+2006 \sqrt{2007}}$ $=\frac{1}{2007 \sqrt{2006}+2006 \sqrt{2007}}\cdot\frac{2007 \sqrt{2006}-2006 \sqrt{2007}}{2007 \sqrt{2006}-2006 \sqrt{2007}}$ $=\frac{2007\sqrt{2006}-2006\sqrt{2007}}{2006 \cdot 2007}$ $Y=\frac{1}{\sqrt{2006}}-\frac{1}{\sqrt{2007}}=\frac{\sqrt{2006}}{2006}-\frac{\sqrt{2007}}{2007}=\frac{2007\sqrt{2006}-2006\sqrt{2007}}{2006 \cdot 2007}$

$X=Y\Rightarrow\mathrm{ C}$

See also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2007_Cyprus_MO/Lyceum/Problem_11&oldid=14868"