A quadratic ratio inequality with a normalized four-variable product

Proved

WorkbookSource.base_8910

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

lean-workbooksource-checked

$a,b,c,d>0$ ,prove that:

$\frac{{a}^{2}+{c}^{2}+{b}^{2}+{d}^{2}}{ab+bc+cd+ad+ac+bd}+27\,{\frac {abcd}{ \left( b+c+d \right) \left( a+c+d \right) \left( d+a+b \right) \left( a+b+c \right) }}\geq 1$

Source: InternLM Lean-Workbook, record lean_workbook_8910 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_8910 (a b c d : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hd : d > 0) : (a^2 + b^2 + c^2 + d^2) / (a * b + b * c + c * d + d * a + a * c + b * d) + 27 * (a * b * c * d) / (b + c + d) / (a + c + d) / (d + a + b) / (a + b + c) ≥ 1  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_8910; Apache-2.0

View graph