A pairwise quadratic reciprocal sum is bounded above

Proved

WorkbookSource.base_4944

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

lean-workbooksource-checked

Let $a,b,c >0$ . Prove: $\frac{{a + b}}{{{a^2} + {b^2}}} + \frac{{b + c}}{{{b^2} + {c^2}}} + \frac{{c + a}}{{{c^2} + {a^2}}} \le \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$

Source: InternLM Lean-Workbook, record lean_workbook_4944 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_4944 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a + b) / (a ^ 2 + b ^ 2) + (b + c) / (b ^ 2 + c ^ 2) + (c + a) / (c ^ 2 + a ^ 2) ≤ 1 / a + 1 / b + 1 / c  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_4944; Apache-2.0

View graph