A sum of mixed quadratic products at fixed sum two

Proved

WorkbookSource.plus_70753

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

lean-workbooksource-checked

Let $a,b,c$ be non-negative real numbers such that $a+b+c=2$ . Prove that $(a^{2}+bc)(b^{2}+ca)+(b^{2}+ca)(c^{2}+ab)+(c^{2}+ab)(a^{2}+bc) \le 3.$

Source: InternLM Lean-Workbook, record lean_workbook_plus_70753 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_70753 (a b c : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (hab : a + b + c = 2) : (a^2 + b * c) * (b^2 + c * a) + (b^2 + c * a) * (c^2 + a * b) + (c^2 + a * b) * (a^2 + b * c) ≤ 3   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_70753; Apache-2.0

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