A cyclic ratio sum with a normalized cubic correction

Proved

WorkbookSource.plus_22367

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

lean-workbooksource-checked

Let $a,b,c$ are positive real numbers. Prove $4 (\frac{a}{b}+\frac{b}{c}+\frac{c}{a})+\frac{9abc}{a^3+b^3+c^3+2(ab^2+bc^2+ca^2)}\ge 13$

Source: InternLM Lean-Workbook, record lean_workbook_plus_22367 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_22367 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : 4 * (a / b + b / c + c / a) + (9 * a * b * c) / (a ^ 3 + b ^ 3 + c ^ 3 + 2 * (a * b ^ 2 + b * c ^ 2 + c * a ^ 2)) ≥ 13   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_22367; Apache-2.0

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