AoPS - Problem

For a non-empty finite complex number set

$A$ , define the "Tao root" of

$A$ as

$\left|\sum_{z\in A} z \right|$ . Given the integer

$n\ge 3$ , let the set

$U_n = \{\cos\frac{2k \pi}{n}+ i\sin\frac{2k \pi}{n}|k=0,1,...,n-1\}.$ Let

$a_n$ be the number of non-empty subsets in which the Tao root of

$U_n$ is

$0$ ,

$b_n$ is the number of non-empty subsets of

$U_n$ whose Tao root is

$1$ . Compare the sizes of

$na_n$ and

$2b_n$ .