A cyclic ratio sum with a refined triple-product correction

Proved

WorkbookSource.base_32804

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

lean-workbooksource-checked

If $x, y, z>0$ prove that $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}+\frac{243xyz}{2(x+y+z)^3+27xyz}\ge 6$

Source: InternLM Lean-Workbook, record lean_workbook_32804 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_32804 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (x / y + y / z + z / x + 243 * x * y * z / (2 * (x + y + z) ^ 3 + 27 * x * y * z)) ≥ 6  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_32804; Apache-2.0

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