A quadratic ratio with a cubic reciprocal correction

Proved

WorkbookSource.base_12560

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

lean-workbooksource-checked

If $x,y,z>0$ prove that: $\frac{3(x^2+y^2+z^2)}{xy+yz+zx}+\frac{3xyz}{(x^3+y^3+z^3)}\ge 4$

Source: InternLM Lean-Workbook, record lean_workbook_12560 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_12560 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : 3 * (x ^ 2 + y ^ 2 + z ^ 2) / (x * y + y * z + z * x) + 3 * x * y * z / (x ^ 3 + y ^ 3 + z ^ 3) ≥ 4  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_12560; Apache-2.0

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