A weighted quadratic ratio sum with a symmetric correction

Proved

WorkbookSource.base_11458

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

lean-workbooksource-checked

For $a, b, c>0$ prove that $\frac{a(3a+b+c)}{b^2+c^2}+\frac{b(3b+c+a)}{c^2+a^2}+\frac{c(3c+a+b)}{a^2+b^2}+\frac{ab+bc+ca}{a^2+b^2+c^2}\ge\frac{17}{2}$

Source: InternLM Lean-Workbook, record lean_workbook_11458 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_11458 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a * (3 * a + b + c) / (b ^ 2 + c ^ 2) + b * (3 * b + c + a) / (c ^ 2 + a ^ 2) + c * (3 * c + a + b) / (a ^ 2 + b ^ 2) + (a * b + b * c + c * a) / (a ^ 2 + b ^ 2 + c ^ 2)) ≥ 17 / 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_11458; Apache-2.0

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