A pairwise ratio sum bounded by a symmetric quadratic ratio

Proved

WorkbookSource.base_24491

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

lean-workbooksource-checked

For $x, y, z > 0$ prove that: $\frac{(x+y+z)^2}{2(xy+yz+zx)} \ge \sum_{cyc} \frac{x+y}{x+y+2z} \ \ ; (1)$

Source: InternLM Lean-Workbook, record lean_workbook_24491 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_24491 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (x + y + z) ^ 2 / (2 * (x * y + y * z + z * x)) ≥ (x + y) / (x + y + 2 * z) + (y + z) / (y + z + 2 * x) + (z + x) / (z + x + 2 * y)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_24491; Apache-2.0

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