A refined cubic inequality at fixed sum three

Proved

WorkbookSource.base_50202

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

lean-workbooksource-checked

Prove that $2(a^2+b^2+c^2)+3abc\ge 9+\frac{abc(ab+bc+ca-3abc)}{6(ab+bc+ca)}$ given $a,b,c>0$ and $a+b+c=3$ .

Source: InternLM Lean-Workbook, record lean_workbook_50202 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_50202 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : 2 * (a ^ 2 + b ^ 2 + c ^ 2) + 3 * a * b * c ≥ 9 + (a * b * c * (a * b + b * c + c * a - 3 * a * b * c)) / (6 * (a * b + b * c + c * a))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_50202; Apache-2.0

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