2012 IMO Problems

Problem

Let ${{a}_{2}}, {{a}_{3}}, \cdots, {{a}_{n}}$ be positive real numbers that satisfy ${{a}_{2}}\cdot {{a}_{3}}\cdots {{a}_{n}}=1$ . Prove that $\left(a_2+1\right)^2\cdot \left(a_3+1\right)^3\cdots \left(a_n+1\right)^n\gneq n^n$

Solution

The inequality between arithmetic and geometric mean implies ${{\left( {{a}_{k}}+1 \right)}^{k}}={{\left( {{a}_{k}}+\frac{1}{k-1}+\frac{1}{k-1}+\cdots +\frac{1}{k-1} \right)}^{k}}\ge {{k}^{k}}\cdot {{a}_{k}}\cdot \frac{1}{{{\left( k-1 \right)}^{k-1}}}=\frac{{{k}^{k}}}{{{\left( k-1 \right)}^{k-1}}}\cdot {{a}_{k}}$ The inequality is strict unless $a_k=\frac1{k-1}$ . Multiplying analogous inequalities for $k=2,\text{ 3, }\cdots \text{, }n$ yields $\left(a_2+1\right)^2\cdot \left(a_3+1\right)^3\cdots \left(a_n+n\right)^n\gneq \frac{2^2}{1^1}\cdot\frac{3^3}{2^2}\cdot \frac{4^4}{3^3}\cdots \frac{n^n}{(n-1)^{n-1}}\cdot a_2a_3\cdots a_n=n^n$

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2012 IMO Problems

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