A cyclic mixed quadratic ratio sum is at least three

Proved

WorkbookSource.base_8649

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

lean-workbooksource-checked

Let a,b,c>0. Prove that $\frac{{{a^2} + bc}}{{b\left( {c + a} \right)}} + \frac{{{b^2} + ca}}{{c\left( {a + b} \right)}} + \frac{{{c^2} + ab}}{{a\left( {b + c} \right)}} \ge 3$

Source: InternLM Lean-Workbook, record lean_workbook_8649 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_8649 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 + b * c) / (b * (c + a)) + (b^2 + c * a) / (c * (a + b)) + (c^2 + a * b) / (a * (b + c)) ≥ 3  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_8649; Apache-2.0

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