A cyclic cubic ratio bounds a symmetric quadratic expression

Proved

WorkbookSource.base_23529

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

lean-workbooksource-checked

Let $a,b,c$ be three positive real numbers . Prove that :

\frac{a^3}{a^2+bc}+\frac{b^3}{b^2+ca}+\frac{c^3}{c^2+ab} \ge \frac{2(a^2+b^2+c^2)+ab+bc+ca}{2(a+b+c)}.

Source: InternLM Lean-Workbook, record lean_workbook_23529 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_23529 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^3 / (a^2 + b * c) + b^3 / (b^2 + c * a) + c^3 / (c^2 + a * b) ) ≥ (2 * (a^2 + b^2 + c^2) + a * b + b * c + c * a) / (2 * (a + b + c))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_23529; Apache-2.0

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