A cyclic product-reciprocal sum bounds the reciprocal squared total

Proved

WorkbookSource.base_10174

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

lean-workbooksource-checked

For $a,b,c>0$ . Prove that: $\sum_{cyc}\frac{1}{a(a+b)}\ge \frac{27}{2(a+b+c)^2}$

Source: InternLM Lean-Workbook, record lean_workbook_10174 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_10174 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (1 / (a * (a + b)) + 1 / (b * (b + c)) + 1 / (c * (c + a))) ≥ 27 / 2 / (a + b + c) ^ 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_10174; Apache-2.0

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