Prove2Me

A rearrangement inequality for cyclic squared ratios

Proved
WorkbookSource.base_15569

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

lean-workbooksource-checked

Prove or disprove that a2+3b2(b+c)2+b2+3c2(c+a)2+c2+3a2(a+b)2≤4∑a2(b+c)2\frac{a^{2}+3b^{2}}{(b+c)^{2}}+\frac{b^{2}+3c^{2}}{(c+a)^{2}}+\frac{c^{2}+3a^{2}}{(a+b)^{2}} \leq 4\sum \frac{a^{2}}{(b+c)^{2}} given a,b,c>0a,b,c>0 using Rearrangement inequality.

Source: InternLM Lean-Workbook, record lean_workbook_15569 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_15569 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 + 3 * b^2) / (b + c)^2 + (b^2 + 3 * c^2) / (c + a)^2 + (c^2 + 3 * a^2) / (a + b)^2 ≤ 4 * (a^2 / (b + c)^2 + b^2 / (c + a)^2 + c^2 / (a + b)^2)  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_15569; Apache-2.0

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