A shifted cyclic cubic ratio with a pair-product correction

Proved

WorkbookSource.base_57027

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

lean-workbooksource-checked

Let $x,y,z$ be positive reals satisfy $x+y+z=3$ Prove that $\frac{x^3}{y^2+2}+\frac{y^3}{z^2+2}+\frac{z^3}{x^2+2}+\Big ( xy+yz+zx \Big ) \geq 4$

Source: InternLM Lean-Workbook, record lean_workbook_57027 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_57027 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) (h : x + y + z = 3) : (x ^ 3 / (y ^ 2 + 2) + y ^ 3 / (z ^ 2 + 2) + z ^ 3 / (x ^ 2 + 2)) + (x * y + y * z + z * x) ≥ 4  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_57027; Apache-2.0

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