A mixed cubic ratio sum bounds twice the total

Proved

WorkbookSource.base_53800

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

lean-workbooksource-checked

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

Source: InternLM Lean-Workbook, record lean_workbook_53800 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_53800 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (b^3 + 2 * a * b * c + c^3) / (a^2 + b * c) + (c^3 + 2 * b * c * a + a^3) / (b^2 + c * a) + (a^3 + 2 * c * a * b + b^3) / (c^2 + a * b) ≥ 2 * (a + b + c)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_53800; Apache-2.0

View graph