1998 USAMO Problems/Problem 3
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.