Difference between revisions of "2006 Cyprus MO/Lyceum/Problem 9"
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==Problem== | ==Problem== | ||
− | 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? |
A. <math>x=y</math> | A. <math>x=y</math> |
Revision as of 11:38, 24 October 2007
Problem
If and , then which of the following is correct?
A.
B.
C.
D.
E. None of these
Solution
The question is asking us for an approximation of the ratio between . 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; , , , 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 |