A shifted quadratic reciprocal sum is at least one third

Proved

WorkbookSource.plus_2307

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

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers such that $a+b+c=3$ . Prove that

\dfrac{1}{2a^2+a+6}+\dfrac{1}{2b^2+b+6}+\dfrac{1}{2c^2+c+6}\geq\dfrac{1}{3}.

Source: InternLM Lean-Workbook, record lean_workbook_plus_2307 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_2307 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : 1 / (2 * a ^ 2 + a + 6) + 1 / (2 * b ^ 2 + b + 6) + 1 / (2 * c ^ 2 + c + 6) ≥ 1 / 3   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_2307; Apache-2.0

View graph