A shifted pairwise reciprocal lower bound

Proved

WorkbookSource.base_42967

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

lean-workbooksource-checked

Let $a,b,c > 0$ , prove that $\frac{1-bc}{b+c}+\frac{1-ca}{c+a}+\frac{1-ab}{a+b} \geq \frac{9 - (a + b + c)^2}{2(a + b + c)}$

Source: InternLM Lean-Workbook, record lean_workbook_42967 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_42967 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (1 - b * c) / (b + c) + (1 - c * a) / (c + a) + (1 - a * b) / (a + b) ≥ (9 - (a + b + c) ^ 2) / (2 * (a + b + c))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_42967; Apache-2.0

View graph