A pairwise ratio sum with a symmetric quadratic correction

Proved

WorkbookSource.plus_10141

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

lean-workbooksource-checked

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

$\frac{a+b}{a+b+2c}+\frac{b+c}{b+c+2a}+\frac{c+a}{c+a+2b}+\frac{2}{3}.\frac{ab+bc+ca}{a^2+b^2+c^2} \leq \frac{13}{6}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_10141 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_10141 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a + b) / (a + b + 2 * c) + (b + c) / (b + c + 2 * a) + (c + a) / (c + a + 2 * b) + (2 / 3) * (a * b + b * c + c * a) / (a ^ 2 + b ^ 2 + c ^ 2) ≤ 13 / 6   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_10141; Apache-2.0

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