MathDB

2

Part of 2023 Bulgaria JBMO TST

Problems(2)

Take the gcd!

Source: Bulgaria JBMO TST 2023, Day 2, Problem 2

6/17/2023
Determine the smallest positive integer n≥2n\geq 2 for which there exists a positive integer mm such that mnmn divides m2023+n2023+nm^{2023} + n^{2023} + n.
number theoryJBMO TSTgreatest common divisorDivisibility
Heavy counting of routes

Source: Bulgaria JBMO TST 2023, Day 3, Problem 2

6/17/2023
On the coast of a circular island there are eight different cities. Initially there are no routes between the cities. We have to construct five straight two-way routes, which do not intersect, so that from each city there are one or two routes. In how many ways can this happen?
countingcombinatorics