MathDB
VJIMC 2019 P4

Source: VJIMC 2019

March 29, 2019
algebrainequalitiesreal analysisVJIMC2019VJIMCVojtech JarnikAnnual Vojtech Jarnic

Problem Statement

Determine the largest constant K≥0K\geq 0 such that aa(b2+c2)(aa−1)2+bb(c2+a2)(bb−1)2+cc(a2+b2)(cc−1)2≥K(a+b+cabc−1)2\frac{a^a(b^2+c^2)}{(a^a-1)^2}+\frac{b^b(c^2+a^2)}{(b^b-1)^2}+\frac{c^c(a^2+b^2)}{(c^c-1)^2}\geq K\left (\frac{a+b+c}{abc-1}\right)^2 holds for all positive real numbers a,b,ca,b,c such that ab+bc+ca=abcab+bc+ca=abc.
Proposed by Orif Ibrogimov (Czech Technical University of Prague).
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