A cyclic cubic ratio sum bounds the total

Proved

WorkbookSource.base_46828

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

lean-workbooksource-checked

Let $a, b, c>0.$ Prove that the following inequality holds: $\frac{a^{3}+3abc}{(b+c)^{2}}+\frac{b^{3}+3abc}{(c+a)^{2}}+\frac{c^{3}+3abc}{(a+b)^{2}}\geq(a+b+c).$

Source: InternLM Lean-Workbook, record lean_workbook_46828 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_46828 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^3 + 3 * a * b * c) / (b + c)^2 + (b^3 + 3 * a * b * c) / (c + a)^2 + (c^3 + 3 * a * b * c) / (a + b)^2 ≥ a + b + c  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_46828; Apache-2.0

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