A cyclic cubic bound under a fixed sum

Proved

WorkbookSource.base_16621

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

lean-workbooksource-checked

Let $a,b,c >0$ & $a+b+c=6$ , $q=ab+bc+ca$ . Prove that:

2(ab^2+bc^2+ca^2)+3abc \le q^2-19q+156

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Check the equality cases, you will find something nice

Source: InternLM Lean-Workbook, record lean_workbook_16621 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_16621 (a b c q : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 6) (hq : q = a * b + b * c + c * a) : 2 * (a * b ^ 2 + b * c ^ 2 + c * a ^ 2) + 3 * a * b * c ≤ q ^ 2 - 19 * q + 156  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_16621; Apache-2.0

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