P6

Part of 2022/2023 Tournament of Towns

Problems(3)

Sum of digits problem

Source: 44th International Tournament of Towns, Junior A-Level P6, Fall 2022

Peter added a positive integer

M{}

to a positive integer

N{}

and noticed that the sum of the digits of the resulting integer is the same as the sum of the digits of

N{}

. Then he added

M{}

to the result again, and so on. Will Peter eventually get a number with the same digit sum as the number

N{}

number theorysum of digitsTournament of Towns

Regular ( or not) tetrahedron

Source: 44th International Tournament of Towns, Senior A-Level P6, Spring 2023

The midpoints of all heights of a certain tetrahedron lie on its inscribed sphere. Is this tetrahedron necessarily regular then?

geometrytetrahedron3D geometry

Parition of set into arithmetic progressions

Source: 44th International Tournament of Towns, Junior A-Level P6, Spring 2023

X{}

be a set of integers which can be partitioned into

N{}

disjoint increasing arithmetic progressions (infinite in both directions), and cannot be partitioned into a smaller number of such progressions. Is such partition into

N{}

progressions unique for every such

X{}

N = 2{}

N = 3

?
Viktor Kleptsyn

number theoryArithmetic Progression