An asymmetric weighted quadratic ratio sum is at least three

Proved

WorkbookSource.base_31921

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

lean-workbooksource-checked

If $a, b, c>0$ , prove: $\begin{aligned}\frac{5a^2}{b^2+3bc+c^2}+\frac{9b^2}{2c^2+5ca+2a^2}+\frac{13c^2}{3a^2+7ab+3b^2}\geq3\end{aligned}$

Source: InternLM Lean-Workbook, record lean_workbook_31921 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_31921 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (5 * a ^ 2 / (b ^ 2 + 3 * b * c + c ^ 2) + 9 * b ^ 2 / (2 * c ^ 2 + 5 * c * a + 2 * a ^ 2) + 13 * c ^ 2 / (3 * a ^ 2 + 7 * a * b + 3 * b ^ 2)) ≥ 3  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_31921; Apache-2.0

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