A cyclic mixed quadratic ratio sum is at least seven halves

Proved

WorkbookSource.base_14789

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

lean-workbooksource-checked

For any three positive reals a, b, c, prove that

$\frac{a^2+3bc}{b^2+c^2}+\frac{b^2+3ca}{c^2+a^2}+\frac{c^2+3ab}{a^2+b^2} \ge \frac{7}{2}$

Source: InternLM Lean-Workbook, record lean_workbook_14789 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_14789 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 + 3 * b * c) / (b^2 + c^2) + (b^2 + 3 * c * a) / (c^2 + a^2) + (c^2 + 3 * a * b) / (a^2 + b^2) ≥ 7 / 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_14789; Apache-2.0

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