MathNet 2025 — Part 02 (100 complete bilingual questions) Question 44

Find the smallest positive integer $k \ge 2$ such that for all $x, y, z \ne 1$ satisfying $x + y + z = 3$ and $xyz = 1$ then the $k$-th root of numbers $|x|, |y|$ and $|z|$ are the side lengths of some triangle.