A squared-product bound with a sum constraint

Proved

WorkbookSource.base_6220

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

lean-workbooksource-checked

Prove that: $x^2y^2z^2+\frac{1}{27}(x+y+z)^3\geq \frac{2}{3}(x+y+z)xyz$ given $x,y,z$ are real numbers such that $x+y+z\geq3$

Source: InternLM Lean-Workbook, record lean_workbook_6220 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_6220 (x y z : ℝ) (h : x + y + z ≥ 3) :
  x^2*y^2*z^2 + 1/27 * (x + y + z)^3 ≥ 2/3 * (x + y + z) * x * y * z  :=  by sorry

Source