A comparison of shifted pairwise ratio sums

Proved

WorkbookSource.base_21189

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

lean-workbooksource-checked

Prove that for positive real numbers $a, b, c$ , the following inequality holds:

\frac{3}{8}+\frac{a}{c+b}+\frac{b}{a+c}+\frac{c}{b+a} \ge \frac{25}{8}\left(\frac{a}{a+2b+2c}+\frac{b}{b+2c+2a}+\frac{c}{c+2a+2b}\right)

Source: InternLM Lean-Workbook, record lean_workbook_21189 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_21189 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : 3 / 8 + a / (c + b) + b / (a + c) + c / (b + a) ≥ 25 / 8 * (a / (a + 2 * b + 2 * c) + b / (b + 2 * c + 2 * a) + c / (c + 2 * a + 2 * b))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_21189; Apache-2.0

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