Difference between revisions of "2012 IMO Problems/Problem 2"
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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> | ||
| + | |||
| + | ==See Also== | ||
| + | |||
| + | {{IMO box|year=2012|num-b=1|num-a=3}} | ||
Latest revision as of 00:23, 19 November 2023
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)
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 | ||
