A cubic ratio and normalized quadratic sum inequality

Proved

WorkbookSource.plus_66050

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

lean-workbooksource-checked

Let $a,b,c > 0$ . Prove that $\frac {(a + b + c)^3}{a^2b + b^2c + c^2a} + \frac {30(a^2 + b^2 + c^2)}{(a + b + c)^2}\ge 19$

Source: InternLM Lean-Workbook, record lean_workbook_plus_66050 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_66050 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a + b + c) ^ 3 / (a ^ 2 * b + b ^ 2 * c + c ^ 2 * a) + 30 * (a ^ 2 + b ^ 2 + c ^ 2) / (a + b + c) ^ 2 ≥ 19   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_66050; Apache-2.0

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