A nested reciprocal sum bounds the pairwise products

Proved

WorkbookSource.base_24505

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

lean-workbooksource-checked

Let $a,b,c > 0$ , prove that $\frac{a}{\frac{1}{a} + \frac{1}{b + c} + \frac{1}{a + b + c}} + \frac{b}{\frac{1}{b} + \frac{1}{c + a} + \frac{1}{a + b + c}} + \frac{c}{\frac{1}{c} + \frac{1}{a + b} + \frac{1}{a + b + c}} \geq \frac{6}{11} \cdot (ab + bc + ca)$

Source: InternLM Lean-Workbook, record lean_workbook_24505 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_24505 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / (1 / a + 1 / (b + c) + 1 / (a + b + c)) + b / (1 / b + 1 / (c + a) + 1 / (a + b + c)) + c / (1 / c + 1 / (a + b) + 1 / (a + b + c))) ≥ (6 / 11) * (a * b + b * c + c * a)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_24505; Apache-2.0

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