A cyclic product ratio bounds a symmetric quadratic expression

Proved

WorkbookSource.base_29412

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

lean-workbooksource-checked

If $a, b, c>0$ prove that $\frac{(a+2b)(b+2c)(c+2a)}{(2a+b+c)(2b+c+a)(2c+a+b)}+\frac{5}{64}\ge\frac{ab+bc+ca}{2(a^2+b^2+c^2)}$

Source: InternLM Lean-Workbook, record lean_workbook_29412 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_29412 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a + 2 * b) * (b + 2 * c) * (c + 2 * a) / (2 * a + b + c) / (2 * b + c + a) / (2 * c + a + b) + 5 / 64 ≥ (a * b + b * c + c * a) / (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_29412; Apache-2.0

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