A pairwise ratio sum with a normalized quadratic correction

Proved

WorkbookSource.base_32906

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

lean-workbooksource-checked

Let $a, b, c>0$ . Prove that $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\frac{3(ab+bc+ca)}{(a+b+c)^2}\ge\frac{5}{2}.$

Source: InternLM Lean-Workbook, record lean_workbook_32906 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_32906 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / (b + c) + b / (c + a) + c / (a + b) + 3 * (a * b + b * c + c * a) / (a + b + c) ^ 2) ≥ 5 / 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_32906; Apache-2.0

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