A cyclic triple-product ratio sum is at most one twelfth of the squared total

Proved

WorkbookSource.base_12764

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

lean-workbooksource-checked

Let $a, b, c, d>0$ . Prove that $\frac{abc}{b+c+d}+\frac{bcd}{c+d+a}+\frac{cda}{d+a+b}+\frac{dab}{a+b+c}\le\frac{(a+b+c+d)^2}{12}$

Source: InternLM Lean-Workbook, record lean_workbook_12764 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_12764 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) : (a * b * c / (b + c + d) + b * c * d / (c + d + a) + c * d * a / (d + a + b) + d * a * b / (a + b + c)) ≤ (a + b + c + d) ^ 2 / 12  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_12764; Apache-2.0

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