Prove2Me

A cyclic linear-over-quadratic reciprocal lower bound

Proved
WorkbookSource.base_11082

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

lean-workbooksource-checked

Let x,y,z positive numbers. Prove that :

xy2+yz+z2+yz2+zx+x2+zx2+xy+y2≥3x+y+z \frac{x}{y^2+yz+z^2}+\frac{y}{z^2+zx+x^2}+\frac{z}{x^2+xy+y^2}\ge \frac{3}{x+y+z}

Pachitariu M.

Source: InternLM Lean-Workbook, record lean_workbook_11082 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_11082 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (x / (y ^ 2 + y * z + z ^ 2) + y / (z ^ 2 + z * x + x ^ 2) + z / (x ^ 2 + x * y + y ^ 2)) ≥ 3 / (x + y + z)  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_11082; Apache-2.0

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