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> | ||
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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 be positive real numbers that satisfy . Prove that
Solution
The inequality between arithmetic and geometric mean implies The inequality is strict unless . Multiplying analogous inequalities for yields
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 |