MathDB

A travelling beetle

Source: 44th International Tournament of Towns, Senior A-Level P4, Fall 2022

2/16/2023

In a checkered square, there is a closed door between any two cells adjacent by side. A beetle starts from some cell and travels through cells, passing through doors; she opens a closed door in the direction she is moving and leaves that door open. Through an open door, the beetle can only pass in the direction the door is opened. Prove that if at any moment the beetle wants to return to the starting cell, it is possible for her to do that.

boardcombinatoricsTournament of Towns

I see an infinite chessboard and I want it painted black...

Source: 44th International Tournament of Towns, Junior A-Level P4, Fall 2022

2/16/2023

Let

n>1

be an integer. A rook stands in one of the cells of an infinite chessboard that is initially all white. Each move of the rook is exactly

n{}

cells in a single direction, either vertically or horizontally, and causes the

n{}

cells passed over by the rook to be painted black. After several such moves, without visiting any cell twice, the rook returns to its starting cell, with the resulting black cells forming a closed path. Prove that the number of white cells inside the black path gives a remainder of

1{}

when divided by

n{}

.

combinatoricsboardTournament of Towns

Integers of three colors

Source: 44th International Tournament of Towns, Junior O-Level P4, Fall 2022

2/16/2023

Is it possible to colour all integers greater than

1{}

in three colours (each integer in one colour, all three colours must be used) so that the colour of the product of any two differently coloured numbers is different from the colour of each of the factors?

number theorycombinatoricsTournament of Towns

Hexagon geo with lengths

Source: 44th International Tournament of Towns, Junior A-Level P4, Spring 2023

1/9/2024

The triangles

AB'C, CA'B

and

BC'A

are constructed on the sides of the equilateral triangle

ABC.

In the resulting hexagon

AB'CA'BC'

each of the angles

\angle A'BC',\angle C'AB'

and

\angle B'CA'

is greater than

120^\circ

and the sides satisfy the equalities

AB' = AC',BC' = BA'

and

CA' = CB'.

Prove that the segments

AB',BC'

and

CA'

can form a triangle.
David Brodsky

geometryhexagonlengths

Ratios of terms of arithmetic progressions

Source: 44th International Tournament of Towns, Senior O-Level P4, Spring 2023

1/9/2024

Let

a_1, a_2, a_3,\ldots

and

b_1, b_2, b_3,\ldots

be infinite increasing arithmetic progressions. Their terms are positive numbers. It is known that the ratio

a_k/b_k

is an integer for all k. Is it true that this ratio does not depend on

k{}

?
Boris Frenkin

number theoryArithmetic Progression

P4

Problems(5)