A pairwise ratio sum with a symmetric product correction

Proved

WorkbookSource.base_32409

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

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers . prove that $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\frac{9abc}{2(a+b+c)(bc+ca+ab)}\ge 2.$

Source: InternLM Lean-Workbook, record lean_workbook_32409 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_32409 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / (b + c) + b / (c + a) + c / (a + b) + (9 * a * b * c) / (2 * (a + b + c) * (b * c + c * a + a * b))) ≥ 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_32409; Apache-2.0

View graph