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ISL 2020 A7

  • Answer

Let nn and kk be positive integers. Prove that for a1,…,an∈[1,2k]a_{1}, \ldots, a_{n} \in\left[1,2^{k}\right] one has ∑i=1naia12+…+ai2⩽4kn\sum_{i=1}^{n} \frac{a_{i}}{\sqrt{a_{1}^{2}+\ldots+a_{i}^{2}}} \leqslant 4 \sqrt{k n} (Iran)

Official text Transcription from NuminaMath olympiads, checked word for word against the official paper. Quoted with source.

Official text: IMO 2020 Shortlist booklet, page 7.

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4 \sqrt{k n}

Solutions

Official solution Solution 1 by the IMO Problem Selection Committee

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