A cyclic quadratic difference ratio sum is at least three halves

Proved

WorkbookSource.base_37861

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

lean-workbooksource-checked

Prove that for $x,y,z \in \mathbf{R^+}$ , $\frac{x(3x-y)}{y(3z+x)}+\frac{y(3y-z)}{z(3x+y)}+\frac{z(3z-x)}{x(3y+z)}\geq \frac{3}{2}$

Source: InternLM Lean-Workbook, record lean_workbook_37861 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_37861 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (x * (3 * x - y) / (y * (3 * z + x)) + y * (3 * y - z) / (z * (3 * x + y)) + z * (3 * z - x) / (x * (3 * y + z))) ≥ 3 / 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_37861; Apache-2.0

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