Difference between revisions of "2012 IMO Problems/Problem 2"

(Created page with "== Problem == Let <math>{{a}_{2}}, {{a}_{3}}, \cdots, {{a}_{n}}</math> be positive real numbers that satisfy <math>{{a}_{2}}\cdot {{a}_{3}}\cdots {{a}_{n}}=1</math> . Prove th...")
 
 
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The inequality is strict unless <math>a_k=\frac1{k-1}</math>. Multiplying analogous inequalities for <math>k=2,\text{ 3, }\cdots \text{, }n</math> yields
 
The inequality is strict unless <math>a_k=\frac1{k-1}</math>. Multiplying analogous inequalities for <math>k=2,\text{ 3, }\cdots \text{, }n</math> yields
 
<cmath>\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</cmath>
 
<cmath>\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</cmath>
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==See Also==
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{{IMO box|year=2012|num-b=1|num-a=3}}

Latest revision as of 01:23, 19 November 2023

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 \[\left(a_2+1\right)^2\cdot \left(a_3+1\right)^3\cdots \left(a_n+n\right)^n\gneq \frac{2^2}{1^1}\cdot\frac{3^3}{2^2}\cdot \frac{4^4}{3^3}\cdots \frac{n^n}{(n-1)^{n-1}}\cdot a_2\cdot a_3\cdots a_n=n^n\]

See Also

2012 IMO (Problems) • Resources
Preceded by
Problem 1
1 2 3 4 5 6 Followed by
Problem 3
All IMO Problems and Solutions