Test Contest 1 Problem A

Let $x, y, z$ be positive real numbers. Show that $$x^2 + xy^2 + xyz^2 \ge 4xyz - 4.$$

import Mathlib.Data.Real.Basic

theorem solution (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) :
    x ^ 2 + x * y ^ 2 + x * y * z ^ 2 ≥ 4 * x * y * z - 4 := sorry

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