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Symmetric inhomogeneous inequality with fractions

Source: Germany 2021, Problem 5

June 16, 2021
inequalitiesinequalities proposedSymmetric inequalityequality case

Problem Statement

a) Determine the largest real number AA with the following property: For all non-negative real numbers x,y,zx,y,z, one has 1+yz1+x2+1+zx1+y2+1+xy1+z2≥A.\frac{1+yz}{1+x^2}+\frac{1+zx}{1+y^2}+\frac{1+xy}{1+z^2} \ge A. b) For this real number AA, find all triples (x,y,z)(x,y,z) of non-negative real numbers for which equality holds in the above inequality.
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