A shifted pairwise-sum-square inequality

Proved

WorkbookSource.base_31229

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

lean-workbooksource-checked

Prove that $227+(ab+bc+ca-23)^{2}\ge 2(a^{2}b+b^{2}c+c^{2}a)+3abc$ given $a,b,c\ge0$ and $a+b+c=9$ .

Source: InternLM Lean-Workbook, record lean_workbook_31229 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_31229 (a b c : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (habc : a + b + c = 9) : 227 + (a * b + b * c + c * a - 23)^2 ≥ 2 * (a^2 * b + b^2 * c + c^2 * a) + 3 * a * b * c  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_31229; Apache-2.0

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