A cyclic weighted quadratic ratio sum is at least one

Proved

WorkbookSource.base_45914

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

lean-workbooksource-checked

Prove that for any $a,b,c>0$ we always have $\frac{a^{2}}{2b^{2}+ca}+\frac{b^{2}}{2c^{2}+ab}+\frac{c^{2}}{2a^{2}+bc}\ge 1$

Source: InternLM Lean-Workbook, record lean_workbook_45914 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_45914 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 / (2 * b^2 + c * a) + b^2 / (2 * c^2 + a * b) + c^2 / (2 * a^2 + b * c)) ≥ 1  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_45914; Apache-2.0

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