MathDB

2

Part of 2019 China Team Selection Test

Problems(4)

Terms in Arithmetic Progression with Equal Product

Source: 2019 China TST Day 1 Q2

3/5/2019
Fix a positive integer n≥3n\geq 3. Does there exist infinitely many sets SS of positive integers {a1,a2,…,an\lbrace a_1,a_2,\ldots, a_n, b1,b2,…,bn}b_1,b_2,\ldots,b_n\rbrace, such that gcd⁡(a1,a2,…,an\gcd (a_1,a_2,\ldots, a_n, b1,b2,…,bn)=1b_1,b_2,\ldots,b_n)=1, {ai}i=1n\lbrace a_i\rbrace _{i=1}^n, {bi}i=1n\lbrace b_i\rbrace _{i=1}^n are arithmetic progressions, and ∏i=1nai=∏i=1nbi\prod_{i=1}^n a_i = \prod_{i=1}^n b_i?
arithmetic sequencenumber theory
Operation on 10 tuple with sum 2019

Source: China TST 2019 Test 2 Day 1 Q2

3/16/2019
Let SS be the set of 1010-tuples of non-negative integers that have sum 20192019. For any tuple in SS, if one of the numbers in the tuple is ≥9\geq 9, then we can subtract 99 from it, and add 11 to the remaining numbers in the tuple. Call thus one operation. If for A,B∈SA,B\in S we can get from AA to BB in finitely many operations, then denote A→BA\rightarrow B.
(1) Find the smallest integer kk, such that if the minimum number in A,B∈SA,B\in S respectively are both ≥k\geq k, then A→BA\rightarrow B implies B→AB\rightarrow A.
(2) For the kk obtained in (1), how many tuples can we pick from SS, such that any two of these tuples A,BA,B that are distinct, A↛BA\not\rightarrow B.
combinatoricsalgorithms
set of positve integers with strange conditions

Source: 2019 China TST Test 3 P2

3/23/2019
Let SS be a set of positive integers, such that n∈Sn \in S if and only if ∑d∣n,d<n,d∈Sd≤n\sum_{d|n,d<n,d \in S} d \le n Find all positive integers n=2k⋅pn=2^k \cdot p where kk is a non-negative integer and pp is an odd prime, such that ∑d∣n,d<n,d∈Sd=n\sum_{d|n,d<n,d \in S} d = n
Divisorsnumber theory
Triangle-free graph with least edges

Source: 2019 China TST Test 4 P2

3/29/2019
A graph G(V,E)G(V,E) is triangle-free, but adding any edges to the graph will form a triangle. It's given that ∣V∣=2019|V|=2019, ∣E∣>2018|E|>2018, find the minimum of ∣E∣|E| .
graph theorycombinatorics