A symmetric ratio sum bounds an asymmetric pairwise expression

Proved

WorkbookSource.plus_25220

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

lean-workbooksource-checked

Let $a, b, c$ be positive real numbers. Prove that $\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}\ge \frac{2a}{b+c}+\frac{b+c}{a+b}+\frac{b+c}{a+c}+3$ .

Source: InternLM Lean-Workbook, record lean_workbook_plus_25220 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_25220 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a + b) / c + (b + c) / a + (c + a) / b ≥ 2 * a / (b + c) + (b + c) / (a + b) + (b + c) / (a + c) + 3   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_25220; Apache-2.0

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