Three quadratic forms bound a mixed cubic product

Proved

WorkbookSource.plus_5556

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

lean-workbooksource-checked

Given $a, b, c \geq\ 0.$ Prove that: $(3a^2-ab+3b^2)(3b^2-bc+3c^2)(3c^2-ca+3a^2) \geq\ \frac{125}{3}abc(a^3+b^3+c^3)$

Source: InternLM Lean-Workbook, record lean_workbook_plus_5556 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_5556 (a b c : ℝ) : (3 * a ^ 2 - a * b + 3 * b ^ 2) * (3 * b ^ 2 - b * c + 3 * c ^ 2) * (3 * c ^ 2 - c * a + 3 * a ^ 2) ≥ 125 / 3 * a * b * c * (a ^ 3 + b ^ 3 + c ^ 3)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_5556; Apache-2.0

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