A degree-eight inequality between symmetric products

Proved

WorkbookSource.base_20269

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

lean-workbooksource-checked

Let $x,y,z> 0$ . Prove or disprove? $(x+y+z)^4(x^2y^2+y^2z^2+z^2x^2)\geq 81x^2y^2z^2(x^2+y^2+z^2)$

Source: InternLM Lean-Workbook, record lean_workbook_20269 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_20269 (x y z : ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) : (x + y + z) ^ 4 * (x ^ 2 * y ^ 2 + y ^ 2 * z ^ 2 + z ^ 2 * x ^ 2) ≥ 81 * x ^ 2 * y ^ 2 * z ^ 2 * (x ^ 2 + y ^ 2 + z ^ 2)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_20269; Apache-2.0

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