A cubic product correction bounds a symmetric quadratic ratio

Proved

WorkbookSource.base_56323

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

lean-workbooksource-checked

Prove that $\frac{9abc}{a^3+b^3+c^3+2(a^2b+b^2c+c^2a)}+2\ge \frac{(a+b+c)^2}{a^2+b^2+c^2}$ for $a,b,c>0$ using elementary methods.

Source: InternLM Lean-Workbook, record lean_workbook_56323 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_56323 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (9 * a * b * c) / (a ^ 3 + b ^ 3 + c ^ 3 + 2 * (a ^ 2 * b + b ^ 2 * c + c ^ 2 * a)) + 2 ≥ (a + b + c) ^ 2 / (a ^ 2 + b ^ 2 + c ^ 2)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_56323; Apache-2.0

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