A four-variable weighted cyclic ratio sum is at least one

Proved

WorkbookSource.base_10806

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

lean-workbooksource-checked

Let $a,b,c,d>0$ . Prove that $\frac{a}{b + 2c + d}+ \frac{b}{c + 2d + a}+ \frac{c}{d + 2a + b}+ \frac{d}{a + 2b + c}\geq1$

Source: InternLM Lean-Workbook, record lean_workbook_10806 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_10806 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) : (a / (b + 2 * c + d) + b / (c + 2 * d + a) + c / (d + 2 * a + b) + d / (a + 2 * b + c)) ≥ 1  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_10806; Apache-2.0

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