A refinement of the cyclic pairwise ratio bound

Proved

WorkbookSource.base_11618

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

lean-workbooksource-checked

For $x, y, z > 0$ real numbers, prove that: $\frac{12(xy+yz+zx)}{(x+y+z)^2} + 2\left( \frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y} \right) \ge 7 \ \ ;$

Greetings from Lorian Saceanu Thanks to my friend BQ for his support!

Source: InternLM Lean-Workbook, record lean_workbook_11618 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_11618 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : 12 * (x * y + y * z + z * x) / (x + y + z) ^ 2 + 2 * (x / (y + z) + y / (x + z) + z / (x + y)) ≥ 7  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_11618; Apache-2.0

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