A weighted cyclic cubic ratio bounds the quadratic mean

Proved

WorkbookSource.base_37005

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

lean-workbooksource-checked

For $a,b,c > 0$ , prove that

$\frac {a^3}{2a + 3b} + \frac {b^3}{2b + 3c} + \frac {c^3}{2c + 3a}\geq \frac {a^2 + b^2 + c^2}{5}$

Source: InternLM Lean-Workbook, record lean_workbook_37005 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_37005 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^3 / (2 * a + 3 * b) + b^3 / (2 * b + 3 * c) + c^3 / (2 * c + 3 * a)) ≥ (a^2 + b^2 + c^2) / 5  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_37005; Apache-2.0

View graph