2022/2023 Tournament of Towns

Part of Tournament Of Towns

Subcontests

(7)

2

It is important to divide pizza equally

N{}

friends and a round pizza. It is allowed to make no more than

100{}

straight cuts without shifting the slices until all cuts are done; then the resulting slices are distributed among all the friends so that each of them gets a share off pizza having the same total area. Is there a cutting which gives the above result if a)

N=201

N=400

Biting chameleons

Chameleons of five colors live on the island. When one chameleon bites another, the color of bitten chameleon changes to one of these five colors according to some rule, and the new color depends only on the color of the bitten and the color of the bitting. It is known that

2023

red chameleons can agree on a sequence of bites between themselves, after which they will all turn blue. What is the smallest

k

that can guarantee that

k

red chameleons, biting only each other, can turn blue? (For example, the rules might be: if a red chameleon bites a green one, the bitten one changes color to blue; if a green one bites a red one, the bitten one remains red, that is, "changes color to red"; if red bites red, the bitten one changes color to yellow, etc. The rules for changing colors may be different.)

3

Sum of digits problem

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{}

Regular ( or not) tetrahedron

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

Parition of set into arithmetic progressions

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

6

5

7

7

8