A cyclic cubic ratio bounds a squared total

Proved

WorkbookSource.base_24661

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

lean-workbooksource-checked

Let $a,b,c>0$ . Prove that: $\frac{a(a^2+bc)}{a+b}+\frac{b(b^2+ac)}{b+c}+\frac{c(c^2+ab)}{c+a} \geq \frac{(a+b+c)^2}{3}$

Source: InternLM Lean-Workbook, record lean_workbook_24661 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_24661 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a * (a ^ 2 + b * c) / (a + b) + b * (b ^ 2 + a * c) / (b + c) + c * (c ^ 2 + a * b) / (c + a)) ≥ (a + b + c) ^ 2 / 3  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_24661; Apache-2.0

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