A symmetric cubic ratio with a quadratic correction

Proved

WorkbookSource.base_4500

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

lean-workbooksource-checked

Let $a, \, b, \, c \, > 0$ . Prove that $\frac{a^3+b^3+c^3+3abc}{(a+b)(b+c)(c+a)}+\frac{(a+b+c)^2}{6(a^2+b^2+c^2)}\ge \frac{5}{4}$

Source: InternLM Lean-Workbook, record lean_workbook_4500 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_4500 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^3 + b^3 + c^3 + 3 * a * b * c) / (a + b) / (b + c) / (c + a) + (a + b + c)^2 / 6 / (a^2 + b^2 + c^2) ≥ 5 / 4  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_4500; Apache-2.0

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