A squared shifted pair-product sum bounds a triple product

Proved

WorkbookSource.plus_67937

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

lean-workbooksource-checked

Prove that $(ab+bc+ca-3)^2 \ge 27(abc-1)$ given $a+b+c=3$ and $a, b, c \in \mathbb{R}$ .

Source: InternLM Lean-Workbook, record lean_workbook_plus_67937 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_67937 (a b c : ℝ) (ha : a + b + c = 3) : (a * b + b * c + c * a - 3) ^ 2 ≥ 27 * (a * b * c - 1)   :=  by sorry

Source