A shifted cyclic reciprocal sum is at least three

Proved

WorkbookSource.plus_17197

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

lean-workbooksource-checked

Let $a,b,c>0$ and $a+b+c=3$ . Prove that $\frac{a+1}{b^2+1}+\frac{b+1}{c^2+1}+\frac{c+1}{a^2+1}\geq3$ .

Source: InternLM Lean-Workbook, record lean_workbook_plus_17197 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_17197 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : (a + 1) / (b ^ 2 + 1) + (b + 1) / (c ^ 2 + 1) + (c + 1) / (a ^ 2 + 1) ≥ 3   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_17197; Apache-2.0

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