A shifted pair-product reciprocal lower bound at fixed sum three

Proved

WorkbookSource.base_23367

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

lean-workbooksource-checked

Let $a,b,c>0$ and $a+b+c =3$ .Prove that ! $\frac {a + b}{2ab + 1} + \frac {b + c}{2bc + 1} + \frac {a + c}{2ac + 1} \ge 2$

Source: InternLM Lean-Workbook, record lean_workbook_23367 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_23367 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : (a + b) / (2 * a * b + 1) + (b + c) / (2 * b * c + 1) + (a + c) / (2 * a * c + 1) ≥ 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_23367; Apache-2.0

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