Difference between revisions of "1998 USAMO Problems/Problem 3"
(Created page with '1998 USAMO Problem #1 Problem: Let <math>a_0,\cdots a_n</math> be real numbers in the interval <math>(0,\frac {\pi}{2})</math> such that <math>\tan{(a_0 - \frac {\pi}{4})} + \t…') |
5849206328x (talk | contribs) m (moved 1998 Problem 1 to 1998 USAMO Problems/Problem 1) |
(No difference)
|
Revision as of 16:59, 11 August 2009
1998 USAMO Problem #1
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.