Difference between revisions of "1999 USAMO Problems/Problem 4"
m (→Resources) |
m (→See Also) |
||
Line 33: | Line 33: | ||
{{USAMO newbox|year=1999|num-b=3|num-a=5}} | {{USAMO newbox|year=1999|num-b=3|num-a=5}} | ||
− | [[Category:Olympiad | + | [[Category:Olympiad Inequality Problems]] |
Revision as of 11:55, 17 September 2012
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 |