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

Proved

WorkbookSource.base_33661

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

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers satisfying $a+b+c=2$ . Prove the inequality $\frac{1}{1+ab}+\frac{1}{1+bc}+\frac{1}{1+ca} \ge \frac{27}{13}$

Source: InternLM Lean-Workbook, record lean_workbook_33661 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_33661 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 2) : 1 / (1 + a * b) + 1 / (1 + b * c) + 1 / (1 + c * a) ≥ 27 / 13  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_33661; Apache-2.0

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