2006 Cyprus MO/Lyceum/Problem 9
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. ! LaTeX Error: \begin{eqnarray*} on input line 20 ended by \end{document}.)
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 | |
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