A reciprocal quadratic sum with a refined pair-product factor

Proved

WorkbookSource.base_55935

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

lean-workbooksource-checked

Prove that $\big(2(ab+bc+ca)-\frac{9abc}{a+b+c}\big) \cdot \big(\frac{1}{a^2+b^2}+\frac{1}{b^2+c^2}+\frac{1}{c^2+a^2}\big) \ge \frac{9}{2}$ where $a,b,c>0$ .

Source: InternLM Lean-Workbook, record lean_workbook_55935 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_55935 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (2 * (a * b + b * c + c * a) - 9 * a * b * c / (a + b + c)) * (1 / (a ^ 2 + b ^ 2) + 1 / (b ^ 2 + c ^ 2) + 1 / (c ^ 2 + a ^ 2)) ≥ 9 / 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_55935; Apache-2.0

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