A mixed quadratic reciprocal sum has a symmetric upper bound

Proved

WorkbookSource.plus_56732

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

lean-workbooksource-checked

Prove that: $\frac{1}{2a^2+bc}+\frac{1}{2b^2+ca}+\frac{1}{2c^2+ab} \le \frac{(a+b+c)^2}{(ab+bc+ca)^2}$ Given: $a,b,c> 0$

Source: InternLM Lean-Workbook, record lean_workbook_plus_56732 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_56732 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (1 / (2 * a ^ 2 + b * c) + 1 / (2 * b ^ 2 + c * a) + 1 / (2 * c ^ 2 + a * b)) ≤ (a + b + c) ^ 2 / (a * b + b * c + c * a) ^ 2   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_56732; Apache-2.0

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