A mixed quadratic ratio sum with a symmetric correction

Proved

WorkbookSource.plus_45975

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

lean-workbooksource-checked

If $a,b,c > 0$ , then

$\frac {a(b + c + 2a)}{a^2 + 2bc} + \frac {b(c + a + 2b)}{b^2 + 2ca} + \frac {c(a + b + 2c)}{c^2 + 2ab}\geq3 + \frac {ab + bc + ca}{a^2 + b^2 + c^2}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_45975 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_45975 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a * (b + c + 2 * a) / (a ^ 2 + 2 * b * c) + b * (c + a + 2 * b) / (b ^ 2 + 2 * c * a) + c * (a + b + 2 * c) / (c ^ 2 + 2 * a * b)) ≥ 3 + (a * b + b * c + c * a) / (a ^ 2 + b ^ 2 + c ^ 2)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_45975; Apache-2.0

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