A cyclic quadratic ratio sum is at least three

Proved

WorkbookSource.base_23519

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

lean-workbooksource-checked

Let $a,b ,c$ be positive real numbers. Prove that $\frac{b^2+c^2}{a^2+ bc}+\frac{c^2+a^2}{b^2+ ca}+\frac{a^2+b^2}{c^2+ ab} \ge 3.$ Proposed by Nguyen Viet Hung

Source: InternLM Lean-Workbook, record lean_workbook_23519 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_23519 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (b^2 + c^2) / (a^2 + b * c) + (c^2 + a^2) / (b^2 + c * a) + (a^2 + b^2) / (c^2 + a * b) ≥ 3  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_23519; Apache-2.0

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