A weighted cyclic quadratic ratio bounds the quadratic mean

Proved

WorkbookSource.base_16261

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

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers. Prove that:

\frac{a^2}{2a+5b}+ \frac{b^2}{2b+5c}+ \frac{c^2}{2c+5a}\geq \frac{3(a^2+b^2+c^2)}{7(a+b+c)}

Source: InternLM Lean-Workbook, record lean_workbook_16261 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_16261 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 / (2 * a + 5 * b) + b^2 / (2 * b + 5 * c) + c^2 / (2 * c + 5 * a)) ≥ (3 * (a^2 + b^2 + c^2)) / (7 * (a + b + c))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_16261; Apache-2.0

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