Problem

Source: Ukraine TST 2012 p12

Tags: inequalities, Subsets, algebra



We shall call the triplet of numbers $a, b, c$ of the interval $[-1,1]$ qualitative if these numbers satisfy the inequality $1 + 2abc\ge a^2 + b^2 + c^2$. Prove that when the triples $a, b, c$, and $x, y, z$ are qualitative, then $ax, by, cz$ is also qualitative.