A sum of triple cubic averages bounds the quadratic sum

Proved

WorkbookSource.base_22008

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

lean-workbooksource-checked

Let $a,b,c,d$ be positive real numbers. Prove that : $\frac{a^{3}+b^{3}+c^{3}}{a+b+c}+\frac{b^{3}+c^{3}+d^{3}}{b+c+d}+\frac{c^{3}+d^{3}+a^{3}}{c+d+a}+\frac{d^{3}+a^{3}+b^{3}}{d+a+b}\geq a^{2}+b^{2}+c^{2}+d^{2}$

Source: InternLM Lean-Workbook, record lean_workbook_22008 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_22008 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) : (a^3 + b^3 + c^3) / (a + b + c) + (b^3 + c^3 + d^3) / (b + c + d) + (c^3 + d^3 + a^3) / (c + d + a) + (d^3 + a^3 + b^3) / (d + a + b) ≥ a^2 + b^2 + c^2 + d^2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_22008; Apache-2.0

View graph