MathDB

Cut nine pentominoes out of a square

Source: 42nd International Tournament of Towns, Junior A-Level P4, Fall 2020

2/18/2023

The

X{}

pentomino consists of five

1\times1

squares where four squares are all adjacent to the fifth one. Is it possible to cut nine such pentominoes from an

8\times 8

chessboard, not necessarily cutting along grid lines? (The picture shows how to cut three such

X{}

pentominoes.)
Alexandr Gribalko

combinatorial geometrySouthern Tournamentcombinatorics

Six segments form two triangles

Source: 42nd International Tournament of Towns, Senior O-Level P4, Fall 2020

2/18/2023

The sides of a triangle are divided by the angle bisectors into two segments each. Is it always possible to form two triangles from the obtained six segments?
Lev Emelyanov

geometryTournament of Towns

Equilateral triangle geo

Source: 42nd International Tournament of Towns, Junior O-Level P4, Fall 2020

2/18/2023

There is an equilateral triangle with side

d{}

and a point

P{}

such that the distances from

P{}

to the vertices of the triangle are positive numbers

a, b, c

. Prove that there exist a point

Q{}

and an equilateral triangle with side

a{}

, such that the distances from

Q{}

to the vertices of this triangle are

b, c, d

.
Alexandr Evnin

geometryTournament of Towns

Human VS cat

Source: 42nd International Tournament of Towns, Senior A-Level P4, Spring 2021

2/18/2023

There is a row of

100N

sandwiches with ham. A boy and his cat play a game. In one action the boy eats the first sandwich from any end of the row. In one action the cat either eats the ham from one sandwich or does nothing. The boy performs 100 actions in each of his turns, and the cat makes only 1 action each turn; the boy starts first. The boy wins if the last sandwich he eats contains ham. Is it true that he can win for any positive integer

N{}

no matter how the cat plays?
Ivan Mitrofanov

combinatoricsgameTournament of Towns

Knights and knaves on foreign island

Source: 42nd International Tournament of Towns, Junior A-Level P4, Spring 2021

2/18/2023

A traveler arrived to an island where 50 natives lived. All the natives stood in a circle and each announced firstly the age of his left neighbour, then the age of his right neighbour. Each native is either a knight who told both numbers correctly or a knave who increased one of the numbers by 1 and decreased the other by 1 (on his choice). Is it always possible after that to establish which of the natives are knights and which are knaves?
Alexandr Gribalko

knights and knavescombinatoricsTournament of Towns

Almost quadratic equation has 100 roots

Source: 42nd International Tournament of Towns, Senior O-Level P4, Spring 2021

2/18/2023

It is well-known that a quadratic equation has no more than 2 roots. Is it possible for the equation

\lfloor x^2\rfloor+px+q=0

with

p\neq 0

to have more than 100 roots?
Alexey Tolpygo

algebrarootsTournament of Towns

Splitting shapes into isosceles triangles

Source: 42nd International Tournament of Towns, Junior O-Level P4, Spring 2021

2/18/2023

[*]Is it possible to split a square into 4 isosceles triangles such that no two are congruent? [*]Is it possible to split an equilateral triangle into 4 isosceles triangles such that no two are congruent?
Vladimir Rastorguev

geometryTournament of Towns

P4

Problems(7)