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2018 JBMO TST- Macedonia, problem 3

Source: 2018 JBMO TST- Macedonia

May 28, 2019
JMMO2018MacedoniaalgebraInequalityinequalities

Problem Statement

Let xx, yy, and zz be positive real numbers such that x+y+z=1x + y + z = 1. Prove that
(x+y)3z+(y+z)3x+(z+x)3y+9xyz≥9(xy+yz+zx)\frac{(x+y)^3}{z} + \frac{(y+z)^3}{x} + \frac{(z+x)^3}{y} + 9xyz \ge 9(xy + yz + zx).
When does equality hold?
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