A weighted quadratic reciprocal lower bound

Proved

WorkbookSource.base_35007

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

lean-workbooksource-checked

Prove that $\frac{1}{a^2+7ab+b^2} + \frac{1}{b^2+7bc+c^2} + \frac{1}{c^2+7ca+a^2} \geqslant \frac{1}{ab+bc+ca}$ when $a,b,c$ is positive real number.

Source: InternLM Lean-Workbook, record lean_workbook_35007 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_35007 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : 1 / (a ^ 2 + 7 * a * b + b ^ 2) + 1 / (b ^ 2 + 7 * b * c + c ^ 2) + 1 / (c ^ 2 + 7 * c * a + a ^ 2) ≥ 1 / (a * b + b * c + c * a)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_35007; Apache-2.0

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