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(k+l+m)^2>= 3 (kl+lm+mk), a >= 3b^2 if a(x+y+z)=b(xy+yz+zx)=xyz

Source: Greece Junior Math Olympiad 2024 p1

March 2, 2024
algebrainequalities

Problem Statement

a) Prove that for all real numbers k,l,mk,l,m holds : (k+l+m)2≥3(kl+lm+mk)(k+l+m)^2 \ge 3 (kl+lm+mk) When does equality holds?
b) If x,y,zx,y,z are positive real numbers and a,ba,b real numbers such that a(x+y+z)=b(xy+yz+zx)=xyz,a(x+y+z)=b(xy+yz+zx)=xyz, prove that a≥3b2a \ge 3b^2. When does equality holds?
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