Difference between revisions of "2006 Cyprus MO/Lyceum/Problem 9"
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If <math>x=\sqrt[3]{4}</math> and <math>y=\sqrt[3]{6}-\sqrt[3]{3}</math>, then which of the following is correct? | If <math>x=\sqrt[3]{4}</math> and <math>y=\sqrt[3]{6}-\sqrt[3]{3}</math>, then which of the following is correct? | ||
| − | + | <math>\mathrm{(A)}\ x=y\qquad\mathrm{(B)}\ x<y\qquad\mathrm{(C)}\ x=2y\qquad\mathrm{(D)}\ x>2y\qquad\mathrm{(E)}\ \text{None of these}</math> | |
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| − | B | ||
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| − | C | ||
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| − | D | ||
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==Solution== | ==Solution== | ||
| − | 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 | + | 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]]. |
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| + | By [[difference of cube]]s, | ||
| + | <cmath> | ||
| + | \begin{align*}\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\end{align*} | ||
| + | </cmath> | ||
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>. | 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>. | ||
Latest revision as of 09:40, 27 April 2008
Problem
If ![$x=\sqrt[3]{4}$](http://latex.artofproblemsolving.com/c/c/2/cc27e42aa8ca74f047f8a090afeef7b57eb84829.png)
and ![$y=\sqrt[3]{6}-\sqrt[3]{3}$](http://latex.artofproblemsolving.com/6/1/8/61891fe4ee8185b4fd6dd052bcd60a6b5ef960e6.png)
, then which of the following is correct?
Solution
The question is asking us for an approximation of the ratio between 
. Thus, we are allowed to multiply both sides by a constant.
By difference of cubes,
![\begin{align*}\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\end{align*}](http://latex.artofproblemsolving.com/0/4/1/0419328786f0df4f4788702e695ca9ff07fae312.png)
We can approximate the terms on the LHS; ![$2\sqrt[3]{18} > 4$](http://latex.artofproblemsolving.com/3/e/b/3eb293723f1af999b2820ff9cd289aa04ef44dcb.png)
, ![$2\sqrt[3]{9} > 4$](http://latex.artofproblemsolving.com/a/1/4/a14b9e0a27fb90ea743567b0fbf75de14f398ccb.png)
, ![$\sqrt[3]{36} > 3$](http://latex.artofproblemsolving.com/0/c/0/0c0485765957d6082fd33d1c7d0cf3c6d432b85d.png)
, so the sum on the left side 
. Hence 
, and the answer is 
.
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 | ||
