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f(x^2)f(y^2) is at least f(xy)^2 for quadratic f

Source: Greece MO 1999

May 27, 2011

quadraticsinequalities proposedinequalities

Problem Statement

Let

f(x)=ax^2+bx+c

, where

a,b,c

are nonnegative real numbers, not all equal to zero. Prove that

f(xy)^2\le f(x^2)f(y^2)

for all real numbers

x,y