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Problem 1199

AIME late

Number theory Difficulty 5.2 Prove it Second Round, March · Netherlands · 2019

If you try to divide the number $19$ by $5$ , you will get a remainder. The number $5$ fits $3$ times in $19$ and you will be left with $4$ as remainder. There are two positive integers $n$ having the following property: if you divide $n^2$ by $2n + 1$ , you will get a remainder of $1000$ . What are these two integers?

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

$666$ and $1999$

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.