MathLabs

Problem 2

Let a,b,ca,b,c be positive real numbers with abc=1abc=1. Prove that 1a3(b+c)+1b3(c+a)+1c3(a+b)≥32\frac1{a^3(b+c)}+\frac1{b^3(c+a)}+\frac1{c^3(a+b)}\ge\frac32.
Step 1 of 3: Invert the variables
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In plain words

The condition abc=1abc=1 remains a product condition after inversion.

a=1/x, b=1/y, c=1/z,xyz=1a=1/x,\ b=1/y,\ c=1/z,\quad xyz=1
Detailed analysis

Put a=1/x, b=1/y, c=1/za=1/x,\ b=1/y,\ c=1/z. Then xyz=1xyz=1, and each cyclic term becomes x2y+z\frac{x^2}{y+z}; write the whole left side as EE.