A mixed quadratic ratio sum is at least twenty-two thirds

Proved

WorkbookSource.base_55716

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

lean-workbooksource-checked

Given $a,b,c>0$ prove that $\frac{a^{2}+16bc}{b^{2}+bc+c^{2}}+\frac{b^{2}+16ca}{c^{2}+ca+a^{2}}+\frac{c^{2}+16ab}{a^{2}+ab+b^{2}}\ge \frac{22}{3}$

Source: InternLM Lean-Workbook, record lean_workbook_55716 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_55716 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 + 16 * b * c) / (b^2 + b * c + c^2) + (b^2 + 16 * c * a) / (c^2 + c * a + a^2) + (c^2 + 16 * a * b) / (a^2 + a * b + b^2) ≥ 22 / 3  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_55716; Apache-2.0

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