MathDB

1

Part of 2021 Korea Winter Program Practice Test

Problems(2)

number theory

Source: 2021 Korea Winter Program Practice Test

2/8/2021
Does there exist such infinite set of positive integers SS that satisfies the condition below?
*for all a,ba,b in SS, there exists an odd integer kk that aa divides bk+1b^k+1.
number theoryIMO Shortlist
A nice trip

Source: 2021 Korea Winter Program Test2 Day1 #1

2/13/2021
There is a group of more than three airports. For any two airports A,BA, B belonging to this group, if there is an aircraft from AA to BB, there is an aircraft from BB to AA. For a list of different airports A0,A1,...AnA_0,A_1,...A_n, define this list as a '[color=#00f]route' if there is an aircraft from AiA_i to Ai+1A_{i+1} for each i=0,1,...,n−1i=0,1,...,n-1. Also, define the beginning of this [color=#00f]route as A0A_0, the end as AnA_n, and the length as nn. (n∈Nn\in \mathbb N) Now, let's say that for any three different pairs of airports (A,B,C)(A,B,C), there is always a [color=#00f]route PP that satisfies the following condition.
Condition: PP begins with AA and ends with BB, and does not include CC.
When the length of the longest of the existing [color=#00f]routes is MM (≥2\ge 2), prove that any two [color=#00f]routes of length MM contain at least two different airports simultaneously.
combinatoricsgraph theory