A quadratic sum times cyclic reciprocals is at least nine halves

Proved

WorkbookSource.plus_8965

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

lean-workbooksource-checked

Prove that for any positive real numbers a,b,c we have the following inequality: $(a^{2}+b^{2}+c^{2})(\frac{1}{a^{2}+ab}+\frac{1}{b^{2}+bc} + \frac{1}{c^{2}+ca} )\geq \frac{9}{2}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_8965 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_8965 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 + b^2 + c^2) * (1 / (a^2 + a * b) + 1 / (b^2 + b * c) + 1 / (c^2 + c * a)) ≥ 9 / 2   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_8965; Apache-2.0

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