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Let a_{1},a_{2},\ldots ,a_{n} be positive real numbers such that a_{1}+a_{2}+\cdots +a_{n}<1. Prove that

\frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_1)(1 - a_2) \cdots (1 - a_n)} \leqslant \frac{1}{n^{n+1}}

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