Difference between revisions of "1998 USAMO Problems/Problem 3"
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Revision as of 08:03, 13 September 2012
Problem
Let be real numbers in the interval such that Prove that .
Solution
Let , where . Then we have
By AM-GM,
- $\prod_{i = 0}^n{\frac {1 + y_i}{n}}\ge \prod_{i = 0}^n{\prod_{j\neq i}{(1 - y_j)^{\frac {1}{n}}}$ (Error compiling LaTeX. Unknown error_msg)
Note that by the addition formula for tangents, .
So , as desired.
See Also
1998 USAMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |