Prove2Me

A cyclic product ratio with a symmetric rational correction

Proved
WorkbookSource.base_54197

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

lean-workbooksource-checked

Let x,y,zx, y, z be positive reals. Prove that: xyz+yzx+zxy+9xyz(x+y+z)2(xy+yz+zx)2≥4(x+y+z)\frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}+9\frac{xyz(x+y+z)^2}{(xy+yz+zx)^2}\ge 4(x+y+z)

I think it's my most nice one so far, I hope to see classical solutions

Source: InternLM Lean-Workbook, record lean_workbook_54197 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_54197 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (x * y / z + y * z / x + z * x / y + 9 * (x * y * z * (x + y + z) ^ 2) / (x * y + y * z + z * x) ^ 2) ≥ 4 * (x + y + z)  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_54197; Apache-2.0

View graph