Challenge 31 : Majoration d’une somme de carrés parfaits

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Soit k\in\mathbb{N}^{\star} et soient n_{1},\cdots,n_{k}\in\mathbb{N}^{\star}.

On note {\displaystyle n=\sum_{i=1}^{k}n_{i}}. Montrer que :

    \[ \sum_{i=1}^{k}n_{i}^{2}\leqslant\left(n-k+1\right)^{2}+k-1\]

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