A shifted reciprocal sum bounded by a normalized square

Proved

WorkbookSource.base_13763

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

lean-workbooksource-checked

Let x,y,z be positive reals. Prove that $\frac {x}{x + 2} + \frac {y}{y + 2} + \frac {z}{z + 2} \ge \frac {(x + y + z)^2}{x^2 + y^2 + z^2 + 2x + 2y + 2z}$

Source: InternLM Lean-Workbook, record lean_workbook_13763 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_13763 (x y z : ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) : (x / (x + 2) + y / (y + 2) + z / (z + 2)) ≥ (x + y + z) ^ 2 / (x ^ 2 + y ^ 2 + z ^ 2 + 2 * x + 2 * y + 2 * z)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_13763; Apache-2.0

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