A product of mixed cyclic ratio sums is at least eight

Proved

WorkbookSource.base_21934

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

lean-workbooksource-checked

prove that $\left( {\frac {x}{x+y}}+3\,{\frac {y}{z+x}} \right) \left( {\frac {y}{y+z}}+3\,{\frac {z}{x+y}} \right) \left( {\frac {z}{z+x}}+3\,{\frac {x}{y+z}} \right) \geq 8$ where $x,y,z>0$

Source: InternLM Lean-Workbook, record lean_workbook_21934 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_21934 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (x / (x + y) + 3 * (y / (z + x))) * (y / (y + z) + 3 * (z / (x + y))) * (z / (z + x) + 3 * (x / (y + z))) ≥ 8  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_21934; Apache-2.0

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