2

Part of 2023 South East Mathematical Olympiad

Problems(2)

A division of the set of positive integers

Source: 2023 China South-east MO Grade 10 P2

A

is a non-empty subset of positive integers. Let

f(A)=\{abc-b-c+2\vert a,b,c\in A\}

Determine all integers

n

1

so that we can divide the set of positive integers into

A_1, A_2, \dots, A_n

A_i\neq \emptyset (i=1, 2, \dots , n)

\forall 1\le i < j \le n, A_i\cap A_j = \emptyset

\bigcup_{i=1}^{n} A_i=\mathbb{N}^*

\forall 1\le i\le n, f(A_i) \subseteq A_i

complex number sums

Source: 2023 China South-east Mathematical Olympiad Grade 11 P2 CSMO

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

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

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

2b_n

complex numbersalgebra