MathDB

3

Part of 2022 China Team Selection Test

Problems(4)

Moving pieces in a lattice

Source: 2022 China TST, Test 1, P3 (posting for better LaTeX)

3/24/2022
Let a,b,c,p,q,ra, b, c, p, q, r be positive integers with p,q,r≥2p, q, r \ge 2. Denote Q={(x,y,z)∈Z3:0≤x≤a,0≤y≤b,0≤z≤c}.Q=\{(x, y, z)\in \mathbb{Z}^3 : 0 \le x \le a, 0 \le y \le b , 0 \le z \le c \}. Initially, some pieces are put on the each point in QQ, with a total of MM pieces. Then, one can perform the following three types of operations repeatedly: (1) Remove pp pieces on (x,y,z)(x, y, z) and place a piece on (x−1,y,z)(x-1, y, z) ; (2) Remove qq pieces on (x,y,z)(x, y, z) and place a piece on (x,y−1,z)(x, y-1, z) ; (3) Remove rr pieces on (x,y,z)(x, y, z) and place a piece on (x,y,z−1)(x, y, z-1).
Find the smallest positive integer MM such that one can always perform a sequence of operations, making a piece placed on (0,0,0)(0,0,0), no matter how the pieces are distributed initially.
combinatoricsinvariantalgorithm
On bounding the sum of pairwise non-divisible positive integers

Source: 2022 China TST, Test 2, P3

3/28/2022
Let a1,a2,…,ana_1, a_2, \ldots, a_n be nn positive integers that are not divisible by each other, i.e. for any i≠ji \neq j, aia_i is not divisible by aja_j. Show that a1+a2+⋯+an≥1.1n2−2n. a_1+a_2+\cdots+a_n \ge 1.1n^2-2n.
Note: A proof of the inequality when nn is sufficient large will be awarded points depending on your results.
inequalitiesnumber theoryDivisibility
Distant n-tuples under modulo

Source: 2022 China TST, Test 3 P3

4/30/2022
Given a positive integer n≥2n \ge 2. Find all nn-tuples of positive integers (a1,a2,…,an)(a_1,a_2,\ldots,a_n), such that 1<a1≤a2≤a3≤⋯≤an1<a_1 \le a_2 \le a_3 \le \cdots \le a_n, a1a_1 is odd, and (1) M=12n(a1−1)a2a3⋯anM=\frac{1}{2^n}(a_1-1)a_2 a_3 \cdots a_n is a positive integer; (2) One can pick nn-tuples of integers (ki,1,ki,2,…,ki,n)(k_{i,1},k_{i,2},\ldots,k_{i,n}) for i=1,2,…,Mi=1,2,\ldots,M such that for any 1≤i1<i2≤M1 \le i_1 <i_2 \le M, there exists j∈{1,2,…,n}j \in \{1,2,\ldots,n\} such that ki1,j−ki2,j≢0,±1(modaj)k_{i_1,j}-k_{i_2,j} \not\equiv 0, \pm 1 \pmod{a_j}.
modular arithmeticalgebranumber theorycombinatoricsabstract algebra
Novel FE with multiset condition

Source: 2022 China TST, Test 4 P3

4/30/2022
Find all functions f:R→Rf: \mathbb R \to \mathbb R such that for any x,y∈Rx,y \in \mathbb R, the multiset {(f(xf(y)+1),f(yf(x)−1)}\{(f(xf(y)+1),f(yf(x)-1)\} is identical to the multiset {xf(f(y))+1,yf(f(x))−1}\{xf(f(y))+1,yf(f(x))-1\}.
Note: The multiset {a,b}\{a,b\} is identical to the multiset {c,d}\{c,d\} if and only if a=c,b=da=c,b=d or a=d,b=ca=d,b=c.
functional equationalgebra