A rational comparison of symmetric quadratic and cubic expressions

Proved

WorkbookSource.base_5736

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

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers. Prove that $1+\frac{8abc}{(a+b)(b+c)(c+a)}\geq \frac{2(ab+bc+ca)}{a^2+b^2+c^2}.$

Source: InternLM Lean-Workbook, record lean_workbook_5736 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_5736 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : 1 + (8 * a * b * c) / (a + b) / (b + c) / (c + a) ≥ 2 * (a * b + b * c + c * a) / (a ^ 2 + b ^ 2 + c ^ 2)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_5736; Apache-2.0

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