Problem: (2001 Poland Finals – #1) Prove the following inequality:
where are positive reals.
Solution: Like the title says, that triangular number looks really fishy… let’s write it as and pair up the terms on the RHS.
We claim that for . We’ll use our good friend AM-GM to show this; in fact, it is quite simple.
as desired. Sum them up to get our inequality. QED.
Comment: Just a clever little application of AM-GM; apparantly not a strong inequality at all, and the only equality case is for all . Nevertheless, slightly on the tricky side which makes it sufficiently satisfying to solve.
Practice Problem: (2006 Poland Finals – #1) Solve in reals:
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