MathDB

2

Part of 2018 China Team Selection Test

Problems(4)

Existence of AP of interesting integers

Source: 2018 China TST Day 1 Q2

1/2/2018
A number nn is interesting if 2018 divides d(n)d(n) (the number of positive divisors of nn). Determine all positive integers kk such that there exists an infinite arithmetic progression with common difference kk whose terms are all interesting.
number theoryarithmetic seriesnumber of divisorsChina TST
A Function to Partition Numbers

Source: 2018 China Team Selection Test 2 Problem 2

1/8/2018
An integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. For example, 4 can be partitioned in five distinct ways: 4 3 + 1 2 + 2 2 + 1 + 1 1 + 1 + 1 + 1 The number of partitions of n is given by the partition function p(n)p\left ( n \right ). So p(4)=5p\left ( 4 \right ) = 5 . Determine all the positive integers so that p(n)+p(n+4)=p(n+2)+p(n+3)p\left ( n \right )+p\left ( n+4 \right )=p\left ( n+2 \right )+p\left ( n+3 \right ).
functionInteger partitionsnumber theorycombinatorics
A Graph Maximum Problem

Source: 2018 China TST 3 Day 1 Problem 2

3/27/2018
Let GG be a simple graph with 100 vertices such that for each vertice uu, there exists a vertice v∈N(u)v \in N \left ( u \right ) and N \left ( u \right ) \cap N \left ( v \right ) = \o . Try to find the maximal possible number of edges in GG. The N(.) N \left ( . \right ) refers to the neighborhood.
combinatoricsTSTgraph theory
The Minimum of a special Statistical Indicator

Source: 2018 China TST 4 Day 1 Problem 2

3/27/2018
There are 3232 students in the class with 1010 interesting group. Each group contains exactly 1616 students. For each couple of students, the square of the number of the groups which are only involved by just one of the two students is defined as their interests−disparityinterests-disparity. Define SS as the sum of the interests−disparityinterests-disparity of all the couples, (322)(= 496)\binom{32}{2}\left ( =\: 496 \right ) ones in total. Determine the minimal possible value of SS.
combinatoricsTST