A symmetric quadratic reciprocal difference is nonnegative

Proved

WorkbookSource.base_10103

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

lean-workbooksource-checked

Let $a,b,c>0$ ,Prove: $f = \frac {2}{8a^{2} + bc} + \frac {2}{8b^{2} + ca} + \frac {2}{8c^{2} + ab} + \frac {1}{a^2 + b^2 + c^2} - \frac {3}{ab + bc + ca} \geqslant 0.$

Source: InternLM Lean-Workbook, record lean_workbook_10103 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_10103 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : 2 / (8 * a ^ 2 + b * c) + 2 / (8 * b ^ 2 + a * c) + 2 / (8 * c ^ 2 + a * b) + 1 / (a ^ 2 + b ^ 2 + c ^ 2) - 3 / (a * b + b * c + a * c) ≥ 0  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_10103; Apache-2.0

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