A weighted four-variable cyclic linear ratio sum is at least eight

Proved

WorkbookSource.base_1764

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

lean-workbooksource-checked

Let $a$ , $b$ , $c$ and $d$ be positive numbers. Prove that:

\frac{a+3b}{b+c}+\frac{b+3c}{c+d}+\frac{c+3d}{d+a}+\frac{d+3a}{a+b}\geq8.

Source: InternLM Lean-Workbook, record lean_workbook_1764 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_1764 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) : (a + 3 * b) / (b + c) + (b + 3 * c) / (c + d) + (c + 3 * d) / (d + a) + (d + 3 * a) / (a + b) ≥ 8  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_1764; Apache-2.0

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