A shifted quadratic reciprocal sum is at least one

Proved

WorkbookSource.plus_58086

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

lean-workbooksource-checked

If $a,b,c>0$ and $a+b+c=3\;,$ Then prove that $\frac{1}{2a^2+1}+\frac{1}{2b^2+1}+\frac{1}{2c^2+1}\geq 1$

Source: InternLM Lean-Workbook, record lean_workbook_plus_58086 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_58086 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (habc : a + b + c = 3) : 1 ≤ 1 / (2 * a ^ 2 + 1) + 1 / (2 * b ^ 2 + 1) + 1 / (2 * c ^ 2 + 1)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_58086; Apache-2.0

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