A shifted pairwise ratio sum bounds a normalized total

Proved

WorkbookSource.base_34909

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

lean-workbooksource-checked

Let $a,b,c > 0$ , prove that $\frac{b+c}{a+1}+\frac{c+a}{b+1}+\frac{a+b}{c+1} \geq \frac{6(a + b + c)}{a + b + c + 3}$

Source: InternLM Lean-Workbook, record lean_workbook_34909 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_34909 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (b + c) / (a + 1) + (c + a) / (b + 1) + (a + b) / (c + 1) ≥ 6 * (a + b + c) / (a + b + c + 3)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_34909; Apache-2.0

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