A weighted squared ratio sum is at most 25 over two

Proved

WorkbookSource.base_48565

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

lean-workbooksource-checked

Let $a, b,c>0$ . Prove that $\displaystyle\sum\limits_{\text{cyc}}$ $\frac{(3a+b+c)^2}{2a^2+(b+c)^2}$ $\le\frac{25}{2}$

Source: InternLM Lean-Workbook, record lean_workbook_48565 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_48565 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (3 * a + b + c) ^ 2 / (2 * a ^ 2 + (b + c) ^ 2) + (3 * b + c + a) ^ 2 / (2 * b ^ 2 + (c + a) ^ 2) + (3 * c + a + b) ^ 2 / (2 * c ^ 2 + (a + b) ^ 2) ≤ 25 / 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_48565; Apache-2.0

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