MathDB

10

Part of 2013 Irish Math Olympiad

Problems(1)

Irish Mathematical Olympiad 2013 (2)

Source: Paper 2 , Problem 10

9/6/2014
Let a,b,ca,b,c be real numbers and let x=a+b+c,y=a2+b2+c2,z=a3+b3+c3x=a+b+c,y=a^2+b^2+c^2,z=a^3+b^3+c^3 and S=2x3−9xy+9z.S=2x^3-9xy+9z . (a)(a) Prove that SS is unchanged when a,b,ca,b,c are replaced by a+t,b+t,c+ta+t,b+t,c+t , respectively , for any real number tt. (b)(b) Prove that (3y−x2)3≥S2. (3y-x^2)^3\ge S^2 .
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