MathDB

For a non-empty finite complex number set

A

, define the "Tao root" of

A

\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

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

Problem Statement