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1

Part of 2018 Regional Competition For Advanced Students

Problems(1)

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

5/28/2018
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
OlympiadAustriaBPSQArithmetic Mean-Geometric Meaninequalities proposedInequalityinequalities