A weighted product ratio sum bounds the total

Proved

WorkbookSource.plus_4154

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

lean-workbooksource-checked

Prove or disprove, for $x, y, z > 0$ .

$\frac{8}{9} \sum_{cyc} \frac{xy}{z}+ \frac{xyz}{xy+yz+zx} \ge x+y+z \ \ ; (2)$

Source: InternLM Lean-Workbook, record lean_workbook_plus_4154 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_4154 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (8 / 9) * (x * y / z + y * z / x + z * x / y) + (x * y * z) / (x * y + y * z + z * x) ≥ x + y + z   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_4154; Apache-2.0

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