MathDB

4

Part of 2019 China Team Selection Test

Problems(4)

Good integer sequences

Source: China TST Test 1 Day 2 Q4

3/11/2019
Call a sequence of positive integers {an}\{a_n\} good if for any distinct positive integers m,nm,n, one has gcd⁡(m,n)∣am2+an2 and gcd⁡(am,an)∣m2+n2.\gcd(m,n) \mid a_m^2 + a_n^2 \text{ and } \gcd(a_m,a_n) \mid m^2 + n^2. Call a positive integer aa to be kk-good if there exists a good sequence such that ak=aa_k = a. Does there exists a kk such that there are exactly 20192019 kk-good positive integers?
number theoryDivisibilityChina TSTLte
Sum of finite and infinite set

Source: China TST 2019 Test 2 Day 2 Q4

3/11/2019
Does there exist a finite set AA of positive integers of at least two elements and an infinite set BB of positive integers, such that any two distinct elements in A+BA+B are coprime, and for any coprime positive integers m,nm,n, there exists an element xx in A+BA+B satisfying x≡n(modm)x\equiv n \pmod m ?
Here A+B={a+b∣a∈A,b∈B}A+B=\{a+b|a\in A, b\in B\}.
modular arithmeticnumber theoryChina TST
Two-variable function

Source: 2019 China TST Test 3 P4

3/29/2019
Find all functions f:R2→Rf: \mathbb{R}^2 \rightarrow \mathbb{R}, such that 1) f(0,x)f(0,x) is non-decreasing ; 2) for any x,y∈Rx,y \in \mathbb{R}, f(x,y)=f(y,x)f(x,y)=f(y,x) ; 3) for any x,y,z∈Rx,y,z \in \mathbb{R}, (f(x,y)−f(y,z))(f(y,z)−f(z,x))(f(z,x)−f(x,y))=0(f(x,y)-f(y,z))(f(y,z)-f(z,x))(f(z,x)-f(x,y))=0 ; 4) for any x,y,a∈Rx,y,a \in \mathbb{R}, f(x+a,y+a)=f(x,y)+af(x+a,y+a)=f(x,y)+a .
algebrafunctional equationfunction
Pairwise not divided

Source: 2019 China TST Test 4 P4

3/29/2019
Prove that there exist a subset AA of {1,2,⋯ ,2n}\{1,2,\cdots,2^n\} with nn elements, such that for any two different non-empty subset of AA, the sum of elements of one subset doesn't divide another's.
number theory