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Let

a_{1},a_{2},\cdots,a_{n}

be positive real numbers satisfying a_{1} \plus{} a_{2} \plus{} \cdots \plus{} a_{n} \equal{} 1. Prove that \left(a_{1}a_{2} \plus{} a_{2}a_{3} \plus{} \cdots \plus{} a_{n}a_{1}\right)\left(\frac {a_{1}}{a_{2}^2 \plus{} a_{2}} \plus{} \frac {a_{2}}{a_{3}^2 \plus{} a_{3}} \plus{} \cdots \plus{} \frac {a_{n}}{a_{1}^2 \plus{} a_{1}}\right)\ge\frac {n}{n \plus{} 1}

Problem Statement