C. Latvia 2021 Problem 2

Let $x, y, z$ be positive real numbers. Show that $$\frac{x ^ 2 + y ^ 2}{x + y} + \frac{y ^ 2 + z ^ 2}{y + z} + \frac{z ^ 2 + x ^ 2}{z + x} \ge x + y + z.$$

import Mathlib.Data.Real.Basic

theorem solution (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) :
    (x ^ 2 + y ^ 2) / (x + y) + (y ^ 2 + z ^ 2) / (y + z) + (z ^ 2 + x ^ 2) / (z + x)
    ≥ x + y + z := sorry

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