Difference between revisions of "2001 IMO Shortlist Problems/A3"
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Revision as of 18:00, 20 August 2008
Problem
Let be arbitrary real numbers. Prove the inequality
Solution
We prove the following general inequality, for arbitrary positive real :
with equality only when
.
We proceed by induction on . For
, we have trivial equality. Now, suppose our inequality holds for
. Then by inductive hypothesis,
If we let
, then we have
with equality only if
.
By the Cauchy-Schwarz Inequality,
with equality only when
. Since
, our equality cases never coincide, so we have the desired strict inequality for
. Thus our inequality is true by induction. The problem statement therefore follows from setting
.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.