Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1998 USAMO Problems/Problem 3"
m (moved 1998 USAMO Problems/Problem 1 to 1998 USAMO Problems/Problem 3: Wrong problem, I fix'd it)
(No difference)

Revision as of 10:54, 16 April 2011

1998 USAMO Problem #1

Problem:

Let $a_0,\cdots a_n$ be real numbers in the interval $(0,\frac {\pi}{2})$ such that $\tan{(a_0 - \frac {\pi}{4})} + \tan{(a_1 - \frac {\pi}{4})} + \cdots + \tan{(a_n - \frac {\pi}{4})}\ge n - 1$. Prove that $\tan{(a_0)}\tan{(a_1)}\cdots \tan{(a_n)}\ge n^{n + 1}$.

Solution:

Let $y_i = \tan{(a_i - \frac {\pi}{4})}$, where $0\le i\le n$. Then we have

  • $y_0 + y_1 + \cdots + y_n\ge n - 1$
  • $1 + y_i\ge \sum_{j\neq i}{(1 - y_j)}$
  • $\frac {1 + y_i}{n}\ge \frac {1}{n}\sum_{j\neq i}{(1 - y_j)}$

By AM-GM,

  • $\frac {1}{n}\sum_{j\neq i}{(1 - y_j)}\ge \prod_{j\neq i}{(1 - y_j)^{\frac {1}{n}}}$
  • $\frac {1 + y_i}{n}\ge \prod_{j\neq i}{(1 - y_j)^{\frac {1}{n}}}$
  • $\prod_{i = 0}^n{\frac {1 + y_i}{n}}\ge \prod_{i = 0}^n{\prod_{j\neq i}{(1 - y_j)^{\frac {1}{n}}}$ (Error compiling LaTeX. Unknown error_msg)
  • $= \prod_{i = 0}^n{(1 - y_i)}$
  • $\prod_{i = 0}^n{\frac {1 + y_i}{1 - y_i}}\ge \prod_{i = 0}^n{n} = n^{n + 1}$

Note that by the addition formula for tangents, $\tan{(a_i)} = \tan{[(a_i - \frac {\pi}{4}) + \frac {\pi}{4}]} = \frac {1 + \tan{(a_i - \frac {\pi}{4})}}{1 - \tan{(a_i - \frac {\pi}{4})}} = \frac {1 + y_i}{1 - y_i}$.

So $\prod_{i = 0}^n{\frac {1 + y_i}{1 - y_i}} = \tan{(a_0)}\tan{(a_1)}\cdots \tan{(a_n)}\ge n^{n + 1}$, as desired.

$\text{QED}$

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1998_USAMO_Problems/Problem_3&oldid=38127"