Difference between revisions of "2006 Cyprus MO/Lyceum/Problem 9"

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==Solution==
 
==Solution==
{{solution}}
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The question is asking us for an approximation of the [[ratio]] between <math>x : y</math>. Thus we are allowed to multiply both sides by a constant. So (by [[difference of cube]]s)
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<cmath>\begin{eqnarray*}\sqrt[3]{4}(\sqrt[3]{36}+\sqrt[3]{18}+\sqrt[3]{9}) &:& (\sqrt[3]{6}-\sqrt[3]{3})(\sqrt[3]{36}+\sqrt[3]{18}+\sqrt[3]{9})\\
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2\sqrt[3]{18} + 2\sqrt[3]{9} + \sqrt[3]{36} &:& 3</cmath>
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We can approximate the terms on the LHS; <math>2\sqrt[3]{18} > 4</math>, <math>2\sqrt[3]{9} > 4</math>, <math>\sqrt[3]{36} > 3</math>, so the sum on the left side <math>> 11</math>. Hence <math>x > 2y</math>, and the answer is <math>\mathrm{(D)}</math>.
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''Remark'': There doesn't seem to be any direct way to calculate a simple ratio between the two terms, but various variations can involve approximating terms by multiplying by certain quantities.
  
 
==See also==
 
==See also==
 
{{CYMO box|year=2006|l=Lyceum|num-b=8|num-a=10}}
 
{{CYMO box|year=2006|l=Lyceum|num-b=8|num-a=10}}
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[[Category:Introductory Algebra Problems]]

Revision as of 22:20, 19 October 2007

Problem

If $x=\sqrt[3]{4}$ and $y=\sqrt[3]{6}-\sqrt[3]{3}$, then which of the following is correct

A. $x=y$

B. $x<y$

C. $x=2y$

D. $x>2y$

E. None of these

Solution

The question is asking us for an approximation of the ratio between $x : y$. Thus we are allowed to multiply both sides by a constant. So (by difference of cubes)

\begin{eqnarray*}\sqrt[3]{4}(\sqrt[3]{36}+\sqrt[3]{18}+\sqrt[3]{9}) &:& (\sqrt[3]{6}-\sqrt[3]{3})(\sqrt[3]{36}+\sqrt[3]{18}+\sqrt[3]{9})\\
2\sqrt[3]{18} + 2\sqrt[3]{9} + \sqrt[3]{36} &:& 3 (Error compiling LaTeX. Unknown error_msg)

We can approximate the terms on the LHS; $2\sqrt[3]{18} > 4$, $2\sqrt[3]{9} > 4$, $\sqrt[3]{36} > 3$, so the sum on the left side $> 11$. Hence $x > 2y$, and the answer is $\mathrm{(D)}$.

Remark: There doesn't seem to be any direct way to calculate a simple ratio between the two terms, but various variations can involve approximating terms by multiplying by certain quantities.

See also

2006 Cyprus MO, Lyceum (Problems)
Preceded by
Problem 8
Followed by
Problem 10
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