A cyclic cubic inequality when the pairwise sum vanishes

Proved

WorkbookSource.base_33161

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

lean-workbooksource-checked

Let $a,b,c$ such that $ab+bc+ca=0.$ Prove that: $a^2(1-b)+b^2(1-c)+c^2(1-a) \ge abc(a+b+c-3).$

Source: InternLM Lean-Workbook, record lean_workbook_33161 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_33161 (a b c : ℝ) (hab : a * b + b * c + c * a = 0) : a^2 * (1 - b) + b^2 * (1 - c) + c^2 * (1 - a) ≥ a * b * c * (a + b + c - 3)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_33161; Apache-2.0

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