MathDB

1

Nobody has any intention of building a wall!

Maker and Breaker are building a wall. Maker has a supply of green cubical building blocks, and Breaker has a supply of red ones, all of the same size. On the ground, a row of

m

squares has been marked in chalk as place-holders. Maker and Breaker now take turns in placing a block either directly on one of these squares, or on top of another block already in place, in such a way that the height of each column never exceeds

n

. Maker places the first block. Maker bets that he can form a green row, i.e. all

m

blocks at a certain height are green. Breaker bets that he can prevent Maker from achieving this. Determine all pairs

(m,n)

of positive integers for which Maker can make sure he wins the bet.

9

1

Partitioning regular hexagons into congruent polygons

A positive integer

n

is Danish if a regular hexagon can be partitioned into

n

congruent polygons. Prove that there are infinitely many positive integers

n

such that both

n

and

2^n+n

are Danish.

8

1

A limping knight on a chessboard

A chess knight has injured his leg and is limping. He alternates between a normal move and a short move where he moves to any diagonally neighbouring cell. The limping knight moves on a

5 \times 6

cell chessboard starting with a normal move. What is the largest number of moves he can make if he is starting from a cell of his own choice and is not allowed to visit any cell (including the initial cell) more than once?

7

1

Colouring a complete graph

Each edge of a complete graph on

30

vertices is coloured either red or blue. It is allowed to choose a non-monochromatic triangle and change the colour of the two edges of the same colour to make the triangle monochromatic. Prove that by using this operation repeatedly it is possible to make the entire graph monochromatic. (A complete graph is a graph where any two vertices are connected by an edge.)

6

1

Initial conditions for a solitaire-type game

Fifteen stones are placed on a

4 \times 4

board, one in each cell, the remaining cell being empty. Whenever two stones are on neighbouring cells (having a common side), one may jump over the other to the opposite neighbouring cell, provided this cell is empty. The stone jumped over is removed from the board.
For which initial positions of the empty cell is it possible to end up with exactly one stone on the board?

15

1

Maximal number of large angles in an equilateral n-gon

Let

n \ge 3

be an integer. What is the largest possible number of interior angles greater than

180^\circ

in an

n

-gon in the plane, given that the

n

-gon does not intersect itself and all its sides have the same length?

5

1

Functional Equation

Find all functions

f:\mathbb{R}\rightarrow\mathbb{R}

such that

f(x^2y)=f(xy)+yf(f(x)+y)

for all real numbers

x

and

y

.

4

1

Linear form

A linear form in

k

variables is an expression of the form

P(x_1,...,x_k)=a_1x_1+...+a_kx_k

with real constants

a_1,...,a_k

. Prove that there exist a positive integer

n

and linear forms

P_1,...,P_n

in

2017

variables such that the equation

x_1\cdot x_2\cdot ... \cdot x_{2017}=P_1(x_1,...,x_{2017})^{2017}+...+P_n(x_1,...,x_{2017})^{2017}

holds for all real numbers

x_1,...,x_{2017}

.

3

1

Fibonacci numbers represented as sum of integers

Positive integers

x_1,...,x_m

(not necessarily distinct) are written on a blackboard. It is known that each of the numbers

F_1,...,F_{2018}

can be represented as a sum of one or more of the numbers on the blackboard. What is the smallest possible value of

m

? (Here

F_1,...,F_{2018}

are the first

2018

Fibonacci numbers:

F_1=F_2=1, F_{k+1}=F_k+F_{k-1}

for

k>1

.)

2

1

Finite set of real numbers

Does there exist a finite set of real numbers such that their sum equals

2

, the sum of their squares equals

3

, the sum of their cubes equals

4

, ..., and the sum of their ninth powers equals

10

?

1

Sequence

Let

a_0,a_1,a_2,...

be an infinite sequence of real numbers satisfying

\frac{a_{n-1}+a_{n+1}}{2}\geq a_n

for all positive integers

n

. Show that

\frac{a_0+a_{n+1}}{2}\geq \frac{a_1+a_2+...+a_n}{n}

holds for all positive integers

n

.

19

1

Number of carries

For an integer

n\geq 1

let

a(n)

denote the total number of carries which arise when adding

2017

and

n\cdot 2017

. The first few values are given by

a(1)=1

,

a(2)=1

,

a(3)=0

, which can be seen from the following: \begin{align*} 001 &&001 && 000 \\ 2017 &&4034 &&6051 \\ +2017 &&+2017 &&+2017\\ =4034 &&=6051 &&=8068\\ \end{align*} Prove that

a(1)+a(2)+...+a(10^{2017}-1)=10\cdot\frac{10^{2017}-1}{9}.

14

1

Lots of perpendiculars

Let

P

be a point inside the acute angle

\angle BAC

. Suppose that

\angle ABP = \angle ACP = 90^{\circ}

. The points

D

and

E

are on the segments

BA

and

CA

, respectively, such that

BD = BP

and

CP = CE

. The points

F

and

G

are on the segments

AC

and

AB

, respectively, such that

DF

is perpendicular to

AB

and

EG

is perpendicular to

AC

. Show that

PF = PG

.

13

1

Find < BAC given MB = OI

Let

ABC

be a triangle in which

\angle ABC = 60^{\circ}

. Let

I

and

O

be the incentre and circumcentre of

ABC

, respectively. Let

M

be the midpoint of the arc

BC

of the circumcircle of

ABC

, which does not contain the point

A

. Determine

\angle BAC

given that

MB = OI

.

18

1

Permutation and residues

Let

p>3

be a prime and let

a_1,a_2,...,a_{\frac{p-1}{2}}

be a permutation of

1,2,...,\frac{p-1}{2}

. For which

p

is it always possible to determine the sequence

a_1,a_2,...,a_{\frac{p-1}{2}}

if it for all

i,j\in\{1,2,...,\frac{p-1}{2}\}

with

i\not=j

the residue of

a_ia_j

modulo

p

is known?

12

1

XP/YQ is independent of m

Line

\ell

touches circle

S_1

in the point

X

and circle

S_2

in the point

Y

. We draw a line

m

which is parallel to

\ell

and intersects

S_1

in a point

P

and

S_2

in a point

Q

. Prove that the ratio

XP/YQ

does not depend on the choice of

m

.

17

1

Equation with factorial

Determine whether the equation

x^4+y^3=z!+7

has an infinite number of solutions in positive integers.

16

1

Friendship and divisibility

Is it possible for any finite group of people to choose a positive integer

N

and assign a positive integer to each person in the group such that the product of two persons' number is divisible by

N

if and only if they are friends?

20

1

Sum of two squares

Let

S

be the set of all ordered pairs

(a,b)

of integers with

0<2a<2b<2017

such that

a^2+b^2

is a multiple of

2017

. Prove that

\sum_{(a,b)\in S}a=\frac{1}{2}\sum_{(a,b)\in S}b.

Proposed by Uwe Leck, Germany

11

1

Collinearity with orthocenter

Let

H

and

I

be the orthocenter and incenter, respectively, of an acute-angled triangle

ABC

. The circumcircle of the triangle

BCI

intersects the segment

AB

at the point

P

different from

B

. Let

K

be the projection of

H

onto

AI

and

Q

the reflection of

P

in

K

. Show that

B

,

H

and

Q

are collinear.
Proposed by Mads Christensen, Denmark

2017 Baltic Way

Subcontests

Nobody has any intention of building a wall!

Partitioning regular hexagons into congruent polygons

A limping knight on a chessboard

Colouring a complete graph

Initial conditions for a solitaire-type game

Maximal number of large angles in an equilateral n-gon

Functional Equation

Linear form

Fibonacci numbers represented as sum of integers

Finite set of real numbers

Sequence

Number of carries

Lots of perpendiculars

Find < BAC given MB = OI

Permutation and residues

XP/YQ is independent of m

Equation with factorial

Friendship and divisibility

Sum of two squares

Collinearity with orthocenter

2017 Baltic Way

Subcontests

Nobody has any intention of building a wall!

Partitioning regular hexagons into congruent polygons

A limping knight on a chessboard

Colouring a complete graph

Initial conditions for a solitaire-type game

Maximal number of large angles in an equilateral n-gon

Functional Equation

Linear form

Fibonacci numbers represented as sum of integers

Finite set of real numbers

Sequence

Number of carries

Lots of perpendiculars

Find &lt; BAC given MB = OI

Permutation and residues

XP/YQ is independent of m

Equation with factorial

Friendship and divisibility

Sum of two squares

Collinearity with orthocenter

Find < BAC given MB = OI