A cyclic pairwise ratio bound with a symmetric correction

Proved

WorkbookSource.base_2889

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

lean-workbooksource-checked

If a,b,c>0 are real numbers, prove that: $\frac{a+b}{b+c}+\frac{b+c}{c+a}+\frac{c+a}{a+b}+3\cdot\frac{ab+bc+ca}{(a+b+c)^2}\geq4$

Source: InternLM Lean-Workbook, record lean_workbook_2889 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_2889 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a + b) / (b + c) + (b + c) / (c + a) + (c + a) / (a + b) + 3 * (a * b + b * c + c * a) / (a + b + c) ^ 2 ≥ 4  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_2889; Apache-2.0

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