MathDB

4

Part of 2018 China Team Selection Test

Problems(4)

Prove periodic if bounded

Source: China TST 2018 Day 2 Q1

1/2/2018
Functions f,g:Z→Zf,g:\mathbb{Z}\to\mathbb{Z} satisfy f(g(x)+y)=g(f(y)+x)f(g(x)+y)=g(f(y)+x) for any integers x,yx,y. If ff is bounded, prove that gg is periodic.
functionalgebra
Floor function and coprime

Source: 2018 China TST 2 Day 2 Q4

1/9/2018
Let k,Mk, M be positive integers such that k−1k-1 is not squarefree. Prove that there exist a positive real α\alpha, such that ⌊α⋅kn⌋\lfloor \alpha\cdot k^n \rfloor and MM are coprime for any positive integer nn.
number theoryfloor functionfunction
A Set Addition Problem

Source: 2018 China TST 3 Day 2 Problem 4

3/27/2018
Suppose A1,A2,⋯ ,An⊆{1,2,⋯ ,2018}A_1,A_2,\cdots ,A_n \subseteq \left \{ 1,2,\cdots ,2018 \right \} and ∣Ai∣=2,i=1,2,⋯ ,n\left | A_i \right |=2, i=1,2,\cdots ,n, satisfying that Ai+Aj,  1≤i≤j≤n,A_i + A_j, \; 1 \le i \le j \le n , are distinct from each other. A+B={a+b∣a∈A, b∈B}A + B = \left \{ a+b|a\in A,\,b\in B \right \}. Determine the maximal value of nn.
combinatoricsset theory
Power of k residues not in arithmetic progression

Source: China TST 4 2018 Day 2 Q4

7/17/2018
Let pp be a prime and kk be a positive integer. Set SS contains all positive integers aa satisfying 1≤a≤p−11\le a \le p-1, and there exists positive integer xx such that xk≡a(modp)x^k\equiv a \pmod p. Suppose that 3≤∣S∣≤p−23\le |S| \le p-2. Prove that the elements of SS, when arranged in increasing order, does not form an arithmetic progression.
modular arithmeticarithmetic sequencenumber theory