A squared cyclic ratio sum with a symmetric correction

Proved

WorkbookSource.base_36205

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

lean-workbooksource-checked

Let $a,b,c>0$ . Prove that $16\sum\limits_{cyc}\left(\frac{a}{a+b}\right)^2+\frac{ab+bc+ca}{a^2+b^2+c^2}\ge13$ It is stronger than $\left(\frac{a}{a+b}\right)^2+\left(\frac{b}{b+c}\right)^2+\left(\frac{c}{c+a}\right)^2\ge\frac{3}{4}$

Source: InternLM Lean-Workbook, record lean_workbook_36205 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_36205 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : 16 * ((a / (a + b)) ^ 2 + (b / (b + c)) ^ 2 + (c / (c + a)) ^ 2) + (a * b + b * c + c * a) / (a ^ 2 + b ^ 2 + c ^ 2) ≥ 13  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_36205; Apache-2.0

View graph