A mixed quadratic ratio sum is at least one half

Proved

WorkbookSource.base_53057

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

lean-workbooksource-checked

Prove that $\sum \frac{a^{2}}{2a^{2}+(b+c)^{2}}\geq \frac{1}{2}$ for $a,b,c>0$ .

Source: InternLM Lean-Workbook, record lean_workbook_53057 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_53057 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 / (2 * a^2 + (b + c)^2) + b^2 / (2 * b^2 + (c + a)^2) + c^2 / (2 * c^2 + (a + b)^2)) ≥ 1 / 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_53057; Apache-2.0

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