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sum 1/(a_1+ a_2) >1 if a_i^2 /(a_i-1)>S, a_>1, sum a_i=S

Source: 2007 Moldova JBMO TST p2

February 20, 2021
algebrainequalities

Problem Statement

The real numbers a1,a2,a3a_1, a_2, a_3 are greater than 11 and have the sum equal to SS. If for any i=1,2,3i = 1, 2, 3, holds the inequality ai2ai−1>S\frac{a_i^2}{a_i-1}>S , prove the inequality 1a1+a2+1a2+a3+1a3+a1>1\frac{1}{a_1+ a_2}+\frac{1}{a_2+ a_3}+\frac{1}{a_3+ a_1}>1
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