A cyclic cubic-over-quadratic sum bounds the total

Proved

WorkbookSource.base_1485

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

lean-workbooksource-checked

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

Source: InternLM Lean-Workbook, record lean_workbook_1485 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_1485 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) : (a^3 / (a^2 + a * b + b^2) + b^3 / (b^2 + b * c + c^2) + c^3 / (c^2 + c * a + a^2)) ≥ (a + b + c) / 3  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_1485; Apache-2.0

View graph