Prove2Me

A cyclic rational difference sum with a symmetric correction

Proved
WorkbookSource.plus_50085

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

lean-workbooksource-checked

For all positive x,y,zx,y,z prove that:

∑cycx(y−z)(2x+y)2+13⋅x2+y2+z2xy+yz+zx≥13\sum_{cyc}\frac{x(y-z)}{(2x+y)^2} +\frac13 \cdot \frac{x^2+y^2+z^2}{xy+yz+zx} \geq \frac13

Source: InternLM Lean-Workbook, record lean_workbook_plus_50085 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.plus_50085 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (x * (y - z) / (2 * x + y) ^ 2 + y * (z - x) / (2 * y + z) ^ 2 + z * (x - y) / (2 * z + x) ^ 2 + 1 / 3 * (x ^ 2 + y ^ 2 + z ^ 2) / (x * y + y * z + z * x)) ≥ 1 / 3   :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_50085; Apache-2.0

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