Difference between revisions of "2012 IMO Problems/Problem 2"
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Revision as of 07:17, 13 October 2012
Problem
Let 
be positive real numbers that satisfy 
. Prove that
![\[\left(a_2+1\right)^2\cdot \left(a_3+1\right)^3\cdots \left(a_n+1\right)^n\gneq n^n\]](http://latex.artofproblemsolving.com/d/a/9/da96a9bd18bbc8b4ae7cf8a1cf32fd86486bf885.png)
Solution
The inequality between arithmetic and geometric mean implies
![\[{{\left( {{a}_{k}}+1 \right)}^{k}}={{\left( {{a}_{k}}+\frac{1}{k-1}+\frac{1}{k-1}+\cdots +\frac{1}{k-1} \right)}^{k}}\ge {{k}^{k}}\cdot {{a}_{k}}\cdot \frac{1}{{{\left( k-1 \right)}^{k-1}}}=\frac{{{k}^{k}}}{{{\left( k-1 \right)}^{k-1}}}\cdot {{a}_{k}}\]](http://latex.artofproblemsolving.com/6/6/a/66a11ef0e8b0981697eb560e958e5322e000789f.png)
The inequality is strict unless 
. Multiplying analogous inequalities for 
yields
![\[\left(a_2+1\right)^2\cdot \left(a_3+1\right)^3\cdots \left(a_n+n\right)^n\gneq \frac{2^2}{1^1}\cdot\frac{3^3}{2^2}\cdot \frac{4^4}{3^3}\cdots \frac{n^n}{(n-1)^{n-1}}\cdot a_2\cdot a_3\cdots a_n=n^n\]](http://latex.artofproblemsolving.com/a/4/7/a47f6e0b9060c3692bc23683168537812a372489.png)
