KVPY 2016 SB Question 2

Algebra Level 4

For any real number $r$ , let $A_r = \{ e^{i \pi r n} : n \text{ is a positive integer} \}$ be a set of complex numbers.

Which of the following is true?

A_1

is a finite set and

A_{\frac1\pi} , A_{0.3}

are infinite sets

A_1, A_{0.3}

are finite sets and

A_{\frac1\pi}

is an infinite set

A_1, A_{\frac1\pi}, A_{0.3}

are all finite sets

A_1, A_{\frac1\pi}, A_{0.3}

are all infinite sets