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IMO ShortList 1998, algebra problem 1

Source: IMO ShortList 1998, algebra problem 1

October 22, 2004

inequalitiesalgebraIMO Shortlistn-variable inequalityArithmetic Mean-Geometric Mean

Problem Statement

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})} \leq \frac{1}{ n^{n+1}}.