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easy and classical inequality

Source: RMO District 2005, 9th Grade, Problem 1

March 5, 2005
inequalitiesalgebrapolynomialAMCUSA(J)MOUSAMOrearrangement inequality

Problem Statement

a) Prove that if x,y>0x,y>0 then xy2+yx2≥1x+1y. \frac x{y^2} + \frac y{x^2} \geq \frac 1x + \frac 1y. b) Prove that if a,b,ca,b,c are positive real numbers, then a+bc2+b+ca2+c+ab2≥2(1a+1b+1c). \frac {a+b}{c^2} + \frac {b+c}{a^2} + \frac {c+a}{b^2} \geq 2 \left( \frac 1a + \frac 1b + \frac 1c \right).
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