A weighted pair-product reciprocal lower bound

Proved

WorkbookSource.base_32454

by wamlart · Sep 13, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

Let $a,b,c>0$ .Prove that :

\frac{1}{(2a+b)(2a+c)}+\frac{1}{(2b+c)(2b+a)}+\frac{1}{(2c+a)(2c+b)}\ge \frac{1}{ab+bc+ca}

Source: InternLM Lean-Workbook, record lean_workbook_32454 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_32454 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : 1 / ((2 * a + b) * (2 * a + c)) + 1 / ((2 * b + c) * (2 * b + a)) + 1 / ((2 * c + a) * (2 * c + b)) ≥ 1 / (a * b + b * c + c * a)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_32454; Apache-2.0

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