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sum ka_k <= {n \choose 2} + sum a_k^k

Source: Switzerland - 2005 Swiss MO Final Round p3

December 26, 2022
inequalitiesalgebra

Problem Statement

Prove for all a1,...,an>0a_1, ..., a_n > 0 the following inequality and determine all cases in where the equaloty holds: ∑k=1nkak≤(n2)+∑k=1nakk.\sum_{k=1}^{n}ka_k\le {n \choose 2}+\sum_{k=1}^{n}a_k^k.
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