A shifted cyclic product ratio bound at fixed sum three

Proved

WorkbookSource.base_9181

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

lean-workbooksource-checked

prove that: $\frac{xy}{y+1}+\frac{yz}{z+1}+\frac{zx}{x+1}\geq \frac{3}{2}xyz$ given $x,y,z>0$ and $x+y+z=3$

Source: InternLM Lean-Workbook, record lean_workbook_9181 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_9181 (x y z : ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) (hx1 : x + y + z = 3) : (x * y) / (y + 1) + (y * z) / (z + 1) + (z * x) / (x + 1) ≥ (3 * x * y * z) / 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_9181; Apache-2.0

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