Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.4 AIME, harder Prove it

1. Let $\left\{a_{n}\right\}$ be a convex sequence, $S_{n}=\sum_{i=1}^{n} a_{i}$ . Prove that for $k<m<n$ , we have
$\frac{n-m}{k} S_{k}+\frac{m-k}{n} S_{n}+\frac{k-n}{m} S_{m} \geqslant 0 .$

Solution

Hint: Use property 6 and property 3.

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