A pairwise ratio sum with a symmetric quadratic upper correction

Proved

WorkbookSource.base_42774

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

lean-workbooksource-checked

Let $a$ , $b$ and $c$ be positive numbers. Prove that:

\frac{a+b}{a+b+2c}+\frac{b+c}{2a+b+c}+\frac{c+a}{a+2b+c}+\frac{ab+bc+ca}{2(a^2+b^2+c^2)} \leq2

Source: InternLM Lean-Workbook, record lean_workbook_42774 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_42774 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a + b) / (a + b + 2 * c) + (b + c) / (2 * a + b + c) + (c + a) / (a + 2 * b + c) + (a * b + b * c + c * a) / (2 * (a ^ 2 + b ^ 2 + c ^ 2)) ≤ 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_42774; Apache-2.0

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