Latvian National 2025 Grade 12 Problem 3

For positive real numbers $x$, $y$, $z$ it holds that $x + y + z = 1$. Prove that
$$\frac{1}{xy - z + 2} + \frac{1}{yz - x + 2} + \frac{1}{xz - y + 2} \ge \frac{27}{16}.$$

_Replace_ `sorry` _in the template below with your solution. Mathlib version used by the checker is v4.28.0_.

import Mathlib.Data.Real.Basic

theorem solution (x y z : ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) (h : x + y + z = 1) :
    1 / (x * y - z + 2) + 1 / (y * z - x + 2) + 1 / (x * z - y + 2) ≥ (27 : ℝ) / 16 := sorry

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