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a\leq a_i\leq\frac{1}{a_i}

Source: 5-th Taiwanese Mathematical Olympiad 1996

January 11, 2007

inequalitiesinequalities proposedn-variable inequality

Problem Statement

Let

0<a\leq 1

be a real number and let

a\leq a_{i}\leq\frac{1}{a_{i}}\forall i=\overline{1,1996}

are real numbers. Prove that for any nonnegative real numbers

k_{i}(i=1,2,...,1996)

such that

\sum_{i=1}^{1996}k_{i}=1

we have

(\sum_{i=1}^{1996}k_{i}a_{i})(\sum_{i=1}^{1996}\frac{k_{i}}{a_{i}})\leq (a+\frac{1}{a})^{2}