1999 USAMO Problems/Problem 4
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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.