A symmetric quadratic ratio with a normalized product correction

Proved

WorkbookSource.base_43259

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

lean-workbooksource-checked

a,b,c>0. Prove: $8\\frac{(a+b+c)^{2}}{ab+bc+ac}+27\\frac{(a+b)(b+c)(a+c)}{(a+b+c)^{3}}\geq 32$

Source: InternLM Lean-Workbook, record lean_workbook_43259 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_43259 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : 8 * (a + b + c) ^ 2 / (a * b + b * c + a * c) + 27 * (a + b) * (b + c) * (a + c) / (a + b + c) ^ 3 ≥ 32  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_43259; Apache-2.0

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