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 3 of 5: Complete the square to prove part (i)
Previous stepNext step
a2+b2+c2=(a+b+c)2−2(ab+bc+ca)=(a+b+c−1)2+1≥1a^2+b^2+c^2=(a+b+c)^2-2(ab+bc+ca)=(a+b+c-1)^2+1 \ge 1
Detailed analysis

Let s=a+b+cs=a+b+c. By Step 2, ab+bc+ca=s−1ab+bc+ca=s-1, so a2+b2+c2=s2−2(ab+bc+ca)=s2−2(s−1)=(s−1)2+1≥1,a^2+b^2+c^2=s^2-2(ab+bc+ca)=s^2-2(s-1)=(s-1)^2+1\ge 1, which is exactly the required inequality, since a2+b2+c2a^2+b^2+c^2 is the target sum from Step 1. Equality holds iff s=1s=1, i.e.\ a+b+c=1a+b+c=1, in which case also ab+bc+ca=s−1=0ab+bc+ca=s-1=0.