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If a+b<2, prove that sum 1/(1+a^2) \leq 2/(1+ab)

Source: Austrian Mathematics Olympiad Regional Competition (Qualifying Round) 2018, Problem 1

May 28, 2018
OlympiadAustriaBPSQArithmetic Mean-Geometric Meaninequalities proposedInequalityinequalities

Problem Statement

If a,ba, b are positive reals such that a+b<2a+b<2. Prove that 11+a2+11+b2≤21+ab\frac{1}{1+a^2}+\frac{1}{1+b^2} \le \frac{2}{1+ab} and determine all a,ba, b yielding equality.
Proposed by Gottfried Perz
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