MathLabs

Problem 3

Let x,y,zx,y,z be positive real numbers and let w=xyz3w=\sqrt[3]{xyz}. Prove that (1+x/y)(1+y/z)(1+z/x)≥2+2(x+y+z)/w(1+x/y)(1+y/z)(1+z/x)\ge2+2(x+y+z)/w.
Step 1 of 4: Expand symmetrically
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(1+x/y)(1+y/z)(1+z/x)=(x+y+z)(1/x+1/y+1/z)−1(1+x/y)(1+y/z)(1+z/x)=(x+y+z)(1/x+1/y+1/z)-1
Detailed analysis

Multiplying out and collecting the six ratios gives (1+x/y)(1+y/z)(1+z/x)=(x+y+z)(1/x+1/y+1/z)−1(1+x/y)(1+y/z)(1+z/x)=(x+y+z)(1/x+1/y+1/z)-1.