A weighted pairwise product reciprocal upper bound

Proved

WorkbookSource.base_37822

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

lean-workbooksource-checked

Let $x,y,z>0$ . Prove that :

{\frac {1}{x+y+z}}\geq {\frac {x}{ \left( y+2\,x \right) \left( z+2\,x \right) }}+{\frac {y}{ \left( 2\,y+z \right) \left( 2\,y+x \right) }}+{\frac {z}{ \left( 2\,z+x \right) \left( 2\,z+y \right) }}

Source: InternLM Lean-Workbook, record lean_workbook_37822 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_37822 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : 1 / (x + y + z) ≥ x / ((y + 2 * x) * (z + 2 * x)) + y / ((2 * y + z) * (2 * y + x)) + z / ((2 * z + x) * (2 * z + y))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_37822; Apache-2.0

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