USAJMO 2022 Problem 1

For which positive integers $m$ does there exist an infinite arithmetic sequence of integers $a_1, a_2, . . .$ and an infinite geometric sequence of integers $g_1, g_2, . . .$ satisfying the following properties?

• $a_n - g_n$ is divisible by $m$ for all integers $n \ge 1$ ;
• $a_2 - a_1$ is not divisible by $m$ .

Holden Mui