A symmetric ratio sum bounds a cyclic pairwise ratio sum

Proved

WorkbookSource.base_1827

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

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers. Prove that the following inequality occurs: $\sum\frac{b+c}{a}\geq4\sum_{cyc}\frac{a}{a+b}$

Source: InternLM Lean-Workbook, record lean_workbook_1827 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_1827 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (b + c) / a + (a + c) / b + (a + b) / c ≥ 4 * (a / (a + b) + b / (b + c) + c / (c + a))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_1827; Apache-2.0

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