A cubed pairwise ratio sum bounds the quadratic sum

Proved

WorkbookSource.base_33480

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

lean-workbooksource-checked

If $a,b,c>0$ then $\frac{(a+b)^3}{c}+\frac{(b+c)^3}{a}+\frac{(c+a)^3}{b} \geq 8(a^2+b^2+c^2).$

Source: InternLM Lean-Workbook, record lean_workbook_33480 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_33480 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a + b) ^ 3 / c + (b + c) ^ 3 / a + (c + a) ^ 3 / b ≥ 8 * (a ^ 2 + b ^ 2 + c ^ 2)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_33480; Apache-2.0

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