A pairwise product ratio sum with a normalized symmetric correction

Proved

WorkbookSource.plus_67404

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

lean-workbooksource-checked

Let $a,b,c>0$ , prove that $\sum {\frac{ab}{ {(a+b)^2}}} + \frac{5}{4} \geq \frac{6(ab+bc+ca)}{(a+b+c)^2}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_67404 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_67404 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) : (a * b / (a + b) ^ 2 + b * c / (b + c) ^ 2 + c * a / (c + a) ^ 2 + 5 / 4) ≥ 6 * (a * b + b * c + c * a) / (a + b + c) ^ 2   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_67404; Apache-2.0

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