2001 IMO Shortlist Problems/A3
Let be arbitrary real numbers. Prove the inequality
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 .
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