1999 USAMO Problems/Problem 4
Problem
Let () be real numbers such that Prove that .
Solution
First, suppose all the are positive. Then Suppose, on the other hand, that without loss of generality, with . If we are done, so suppose that . Then , so Since is a positive real for all , it follows that
\[\sum_{i=k+1}^n a_i^2 \le \left( \sum_{i=k+1}^n} -a_i \right)^2 \le (2k-n)^2 .\] (Error compiling LaTeX. Unknown error_msg)
Then Since , . It follows that , as desired.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
1999 USAMO (Problems • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |