City Zhautykov 2023 Grade 8 Problem 2

Show that for all positive reals $a, b, c$, $$\frac{a}{{bc}} + \frac{b}{{ac}} + \frac{c}{{ab}} \ge \frac{2}{a} + \frac{2}{b} - \frac{2}{c}.$$ For which $a, b, c$ does this become an equality?

Replace sorry in the template below with your solution. See Answer Bank for acceptible answer declarations. Mathlib version used by the checker is v4.29.0.

import Mathlib.Data.Real.Basic

def answer : Set (ℝ × ℝ × ℝ) := sorry

theorem solution (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :
    a / (b * c) + b / (a * c) + c / (a * b) ≥ 2 / a + 2 / b - 2 / c ∧
    (a / (b * c) + b / (a * c) + c / (a * b) = 2 / a + 2 / b - 2 / c ↔ (a, b, c) ∈ answer) := sorry

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