A shifted quadratic reciprocal lower bound at fixed sum three

Proved

WorkbookSource.base_21627

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

lean-workbooksource-checked

Prove that for positive reals x, y, z such that x + y + z = 3, the following inequality holds: $\dfrac{x+1}{x^2+2x}+\dfrac{y+1}{y^2+2y}+\dfrac{z+1}{z^2+2z} \ge 2$ .

Source: InternLM Lean-Workbook, record lean_workbook_21627 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_21627 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) (h : x + y + z = 3) : (x + 1) / (x ^ 2 + 2 * x) + (y + 1) / (y ^ 2 + 2 * y) + (z + 1) / (z ^ 2 + 2 * z) ≥ 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_21627; Apache-2.0

View graph