A shifted cubic reciprocal bound at fixed sum three

Proved

WorkbookSource.base_6690

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

lean-workbooksource-checked

Let $x,y,z$ be pozitive real number satisfyng $x+y+z=3$ . Prove that $\sum\limits_{cyclic}{\frac{x^3+4}{x+2}}\ge 5.$

Source: InternLM Lean-Workbook, record lean_workbook_6690 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_6690 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) (h : x + y + z = 3) : (x^3 + 4)/(x + 2) + (y^3 + 4)/(y + 2) + (z^3 + 4)/(z + 2) ≥ 5  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_6690; Apache-2.0

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