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Two inequalities resembling a little to Arquady’s preffered inequality-naming

Source: Romanian District Olympiad 2009, Grade IX, Problem 3

October 7, 2018
inequalities

Problem Statement

a) For a,b≥0 a,b\ge 0 and x,y>0, x,y>0, show that: a3x2+b3y2≥(a+b)3(x+y)2. \frac{a^3}{x^2} +\frac{b^3}{y^2}\ge \frac{(a+b)^3}{(x+y)^2} .
b) For a,b,c≥0 a,b,c\ge 0 and x,y,z>0 x,y,z>0 under the condition a+b+c=x+y+z, a+b+c=x+y+z, prove that: a3x2+b3y2+c3z2≥a+b+c. \frac{a^3}{x^2} +\frac{b^3}{y^2} +\frac{c^3}{z^2} \ge a+b+c.
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