Prove2Me

A cyclic reciprocal ratio bounds a normalized quadratic sum

Proved
WorkbookSource.base_18503

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

lean-workbooksource-checked

For all positive x,y,zx,y,z prove that: ∑cycz2x+y≥13⋅x2+y2+z2xy+yz+zx+23 \sum_{cyc}\frac{z}{2x+y} \geq \frac13 \cdot \frac{x^2+y^2+z^2}{xy+yz+zx}+\frac{2}{3}

Source: InternLM Lean-Workbook, record lean_workbook_18503 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_18503 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (z / (2 * x + y) + x / (2 * y + z) + y / (2 * z + x)) ≥ 1 / 3 * (x ^ 2 + y ^ 2 + z ^ 2) / (x * y + y * z + z * x) + 2 / 3  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_18503; Apache-2.0

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