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 2 of 4: Apply AM-GM to reciprocals
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1/x+1/y+1/z≥3/w1/x+1/y+1/z\ge3/w
Detailed analysis

Since (xyz)1/3=w(xyz)^{1/3}=w, AM-GM gives 1/x+1/y+1/z≥3/(xyz)1/3=3/w1/x+1/y+1/z\ge3/(xyz)^{1/3}=3/w. Therefore the left side is at least 3(x+y+z)/w−13(x+y+z)/w-1.