A pairwise reciprocal lower bound involving the symmetric quadratic sum

Proved

WorkbookSource.base_12167

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

lean-workbooksource-checked

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

\dfrac{1}{a(b+c)}+\dfrac{1}{b(c+a)}+\dfrac{1}{c(a+b)} \ge \frac{9}{2(ab+bc+ca)}

Source: InternLM Lean-Workbook, record lean_workbook_12167 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_12167 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : 1 / (a * (b + c)) + 1 / (b * (c + a)) + 1 / (c * (a + b)) ≥ 9 / (2 * (a * b + b * c + c * a))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_12167; Apache-2.0

View graph