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Let $a_1\geq a_2\geq\ldots\geq a_{100n}$ be positive integers such that if we choose any $2n+1$ numbers from them, the sum of the largest $n$ is strictly greater than the sum of the remaining $n+1$. Prove that

$$(n+1)(a_1+\ldots+a_n)>a_{n+1}+\ldots+a_{100n}.$$

For the case $n=1$, this becomes $a_1\geq\ldots\geq a_{100}$, and we need to prove $2a_1>a_2+\ldots+a_{100}$. We have $a_1>a_2+a_3, a_2>a_3+a_4, \ldots$, and it's not clear how the desired inequality follows.

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$$ 2a_1 > 2a_2+2a_3>a_2+2a_3+a_3+a_4 = a_2+a_3+a_4+2a_3 $$ Similar for $a_3$ and you will gain desired inequality. Then just use induction.

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