A cyclic quadratic ratio sum is at least one

Proved

WorkbookSource.plus_67042

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

lean-workbooksource-checked

If $a,b,c$ are positive reals, prove that $\sum_{cyc}\frac{a^2}{ab+2b^2} \ge 1$ .

Source: InternLM Lean-Workbook, record lean_workbook_plus_67042 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_67042 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 / (a * b + 2 * b^2) + b^2 / (b * c + 2 * c^2) + c^2 / (c * a + 2 * a^2)) ≥ 1   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_67042; Apache-2.0

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