A pairwise ratio sum with a symmetric quadratic correction

Proved

WorkbookSource.base_25364

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

lean-workbooksource-checked

Prove, that all positive real numbers $a$ , $b$ and $c$ satisfy the inequality

\frac{2a}{b+c}+\frac{2b}{c+a}+\frac{2c}{a+b}+\frac{ab+bc+ca}{a^2+b^2+c^2} \geq 4.

Source: InternLM Lean-Workbook, record lean_workbook_25364 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_25364 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (2 * a / (b + c) + 2 * b / (c + a) + 2 * c / (a + b) + (a * b + b * c + c * a) / (a ^ 2 + b ^ 2 + c ^ 2)) ≥ 4  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_25364; Apache-2.0

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