A symmetric rational comparison of pairwise and triple products

Proved

WorkbookSource.base_161

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

lean-workbooksource-checked

For positive reals $a,b,c$ prove: $\frac{a^2 +b^2 +c^2 +4(ab+bc+ca)}{2abc} \geqslant \frac{9(a+b+c)}{2(ab+bc+ca)} +2(\frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{c+a})$

Source: InternLM Lean-Workbook, record lean_workbook_161 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_161 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 + b^2 + c^2 + 4 * (a * b + b * c + c * a)) / (2 * a * b * c) ≥ (9 * (a + b + c)) / (2 * (a * b + b * c + c * a)) + 2 * (1 / (a + b) + 1 / (b + c) + 1 / (c + a))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_161; Apache-2.0

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