Difference between revisions of "2008 IMO Problems/Problem 2"
(New page: == Problem 2 == '''(i)''' If <math>x</math>, <math>y</math> and <math>z</math> are three real numbers, all different from <math>1</math>, such that <math>xyz = 1</math>, then prove that <m...) |
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Revision as of 21:18, 4 September 2008
Problem 2
(i) If , and are three real numbers, all different from , such that , then prove that . (With the sign for cyclic summation, this inequality could be rewritten as .)
(ii) Prove that equality is achieved for infinitely many triples of rational numbers , and .
Solution
Consider the transormation defined by and put . Since maps rational numbers to rational, the problem is equivalent to showing that
given that
\[\frac{\alpha}{\alpha+1)\frac{\beta}{\beta+1) \frac{\gamma}{\gamma+1) = 1 \quad (2)\] (Error compiling LaTeX. Unknown error_msg)
and that the equallity holds for infinitely many triplets of .