A weighted cyclic cubic ratio bounds the quadratic mean

Proved

WorkbookSource.base_18837

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

lean-workbooksource-checked

Prove that for $a, b, c > 0$ , $\frac{a^{3}}{b+2c}+\frac{b^{3}}{c+2a}+\frac{c^{3}}{a+2b}\geq\frac{a^{2}+b^{2}+c^{2}}{3}$

Source: InternLM Lean-Workbook, record lean_workbook_18837 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_18837 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^3 / (b + 2 * c) + b^3 / (c + 2 * a) + c^3 / (a + 2 * b)) ≥ (a^2 + b^2 + c^2) / 3  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_18837; Apache-2.0

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