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Difference between revisions of "2007 Cyprus MO/Lyceum/Problem 11"

 
(→‎Solution: You cannot use calculators in this competition (This was not a nescessary operation))
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==Solution==
 
==Solution==
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<math>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}}{4026042}</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>Y=\frac{1}{\sqrt{2006}}-\frac{1}{\sqrt{2007}}=\frac{\sqrt{2006}}{2006}-\frac{\sqrt{2007}}{2007}=\frac{2007\sqrt{2006}-2006\sqrt{2007}}{4026042}</math>
+
<math>=\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 \times 2007}</math>
  
 
<math>X=Y\Rightarrow\mathrm{ C}</math>
 
<math>X=Y\Rightarrow\mathrm{ C}</math>

Revision as of 12:25, 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

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

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

See also

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