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Sum of third powers equals 3, sum of fifth powers equal 5

Source: Baltic Way 2001

November 17, 2010

inequalities proposedinequalities

Problem Statement

Let

a_1, a_2,\ldots , a_n

be positive real numbers such that

\sum_{i=1}^na_i^3=3

and

\sum_{i=1}^na_i^5=5

. Prove that

\sum_{i=1}^na_i>\frac{3}{2}