Prove2Me

A weighted cyclic cubic ratio bounds one third of the total

Proved
WorkbookSource.base_51012

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

lean-workbooksource-checked

Let aa , bb and cc be positive numbers. Prove that: a(a2+2bc)(b+2c)2+b(b2+2ac)(c+2a)2+c(c2+2ab)(a+2b)2≥a+b+c3\frac{a(a^2+2bc)}{(b+2c)^2}+\frac{b(b^2+2ac)}{(c+2a)^2}+\frac{c(c^2+2ab)}{(a+2b)^2}\geq \frac{a+b+c}{3}

Source: InternLM Lean-Workbook, record lean_workbook_51012 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_51012 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a * (a ^ 2 + 2 * b * c) / (b + 2 * c) ^ 2 + b * (b ^ 2 + 2 * a * c) / (c + 2 * a) ^ 2 + c * (c ^ 2 + 2 * a * b) / (a + 2 * b) ^ 2) ≥ (a + b + c) / 3  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_51012; Apache-2.0

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