A cyclic sum of quadratic ratios has a constant lower bound

Proved

WorkbookSource.base_6600

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

lean-workbooksource-checked

Let a,b,c>0. Prove that: $\frac{{{{\left( {2a + b + c} \right)}^2}}}{{{a^2} + 11bc}} + \frac{{{{\left( {2b + c + a} \right)}^2}}}{{{b^2} + 11ca}} + \frac{{{{\left( {2c + a + b} \right)}^2}}}{{{c^2} + 11ab}} \ge 4$

Source: InternLM Lean-Workbook, record lean_workbook_6600 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_6600 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (2 * a + b + c) ^ 2 / (a ^ 2 + 11 * b * c) + (2 * b + c + a) ^ 2 / (b ^ 2 + 11 * c * a) + (2 * c + a + b) ^ 2 / (c ^ 2 + 11 * a * b) ≥ 4  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_6600; Apache-2.0

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