MathDB

Dissecting a rectangle into similar non-congruent ones

Source: 42nd International Tournament of Towns, Senior A-Level P5, Fall 2020

2/18/2023

Does there exist a rectangle which can be cut into a hundred rectangles such that all of them are similar to the original one but no two are congruent?
Mikhail Murashkin

combinatoricscombinatorial geometryTournament of Towns

Easy NT with cubes

Source: 42nd International Tournament of Towns, Junior A-Level P5, Fall 2020

2/18/2023

Do there exist 100 positive distinct integers such that a cube of one of them equals the sum of the cubes of all the others?
Mikhail Evdokimov

number theoryTournament of Towns

A hundred and one coins in a circle

Source: 42nd International Tournament of Towns, Senior O-Level P5, Fall 2020

2/18/2023

There are 101 coins in a circle, each weights 10g or 11g. Prove that there exists a coin such that the total weight of the

k{}

coins to its left is equal to the total weight of the

k{}

coins to its right where a)

k = 50

and b)

k = 49

.
Alexandr Gribalko

combinatoricsTournament of Towns

Did the elephant go to the gym

Source: 42nd International Tournament of Towns, Junior O-Level P5, Fall 2020

2/18/2023

The director of a Zoo has bought eight elephants numbered by

1, 2, \ldots , 8

. He has forgotten their masses but he remembers that each elephant starting with the third one has the mass equal to the sum of the masses of two preceding ones. Suddenly the director hears a rumor that one of the elephants has lost his mass. How can the director perform two weightings on balancing scales without weights to either find this elephant or make sure that this was just a rumor? (It is known that no elephant gained mass and no more than one elephant lost mass.)
Alexandr Gribalko

combinatoricsTournament of Towns

Switching lightbulbs on

Source: 42nd International Tournament of Towns, Junior A-Level P5, Spring 2021

2/18/2023

In the center of each cell of a checkered rectangle

M{}

there is a point-like light bulb. All the light bulbs are initially switched off. In one turn it is allowed to choose a straight line not intersecting any light bulbs such that on one side of it all the bulbs are switched off, and to switch all of them on. In each turn at least one bulb should be switched on. The task is to switch on all the light bulbs using the largest possible number of turns. What is the maximum number of turns if:
[*]

M

is a square of size

21 \times 21

; [*]

M

is a rectangle of size

20 \times 21

?
Alexandr Shapovalov

Kyiv Tournamentcombinatorics

K lies on the line BM

Source: 42nd International Tournament of Towns, Senior O-Level P5, Spring 2021

2/18/2023

Let

O{}

be the circumcenter of an acute triangle

ABC

. Let

M{}

be the midpoint of

AC

. The straight line

BO

intersects the altitudes

AA_1{}

and

CC_1{}

at the points

H_a

and

H_c

respectively. The circumcircles of the triangles

BH_aA

and

BH_cC

have a second point of intersection

K{}

. Prove that

K{}

lies on the straight line

BM

.
Mikhail Evdokimov

geometryTournament of Towns

Dominoes and a token on a board

Source: 42nd International Tournament of Towns, Junior O-Level P5, Spring 2021

2/18/2023

There are several dominoes on a board such that each domino occupies two adjacent cells and none of the dominoes are adjacent by side or vertex. The bottom left and top right cells of the board are free. A token starts at the bottom left cell and can move to a cell adjacent by side: one step to the right or upwards at each turn. Is it always possible to move from the bottom left to the top right cell without passing through dominoes if the size of the board is a)

100 \times 101

cells and b)

100 \times 100

cells?
Nikolay Chernyatiev

combinatoricsboardTournament of Towns

P5

Problems(7)