A cyclic weighted quadratic ratio sum bounds the total

Proved

WorkbookSource.base_2980

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

lean-workbooksource-checked

Let $a,b,c >0$ .Prove that $\frac{a^{2}+8b^{2}}{4b+5c}+\frac{b^{2}+8c^{2}}{4c+5a}+\frac{c^{2}+8a^{2}}{4a+5b}\geq a+b+c$

Source: InternLM Lean-Workbook, record lean_workbook_2980 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_2980 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 + 8 * b^2) / (4 * b + 5 * c) + (b^2 + 8 * c^2) / (4 * c + 5 * a) + (c^2 + 8 * a^2) / (4 * a + 5 * b) ≥ a + b + c  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_2980; Apache-2.0

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