MathLabs

Problem 2

Let xx, yy, zz be real numbers, all different from 11, such that xyz=1xyz=1. Prove that x2(x−1)2+y2(y−1)2+z2(z−1)2≥1,\frac{x^2}{(x-1)^2}+\frac{y^2}{(y-1)^2}+\frac{z^2}{(z-1)^2}\ge 1, and prove that equality holds for infinitely many triples of rational numbers xx, yy, zz.
Step 2 of 5: Translate xyz = 1 into a linear relation between a+b+c and ab+bc+ca
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xyz=1  ⟺  abc=(a−1)(b−1)(c−1)  ⟺  ab+bc+ca=a+b+c−1xyz=1 \iff abc=(a-1)(b-1)(c-1) \iff ab+bc+ca=a+b+c-1
Detailed analysis

Substituting x=aa−1x=\frac{a}{a-1}, y=bb−1y=\frac{b}{b-1}, z=cc−1z=\frac{c}{c-1} into xyz=1xyz=1 gives abc=(a−1)(b−1)(c−1)abc=(a-1)(b-1)(c-1). Expanding the right side, (a−1)(b−1)(c−1)=abc−(ab+bc+ca)+(a+b+c)−1(a-1)(b-1)(c-1)=abc-(ab+bc+ca)+(a+b+c)-1, so the abcabc terms cancel and we obtain ab+bc+ca=a+b+c−1ab+bc+ca=a+b+c-1.