MathDB

1

Removing successive numbers from an interval.

1

to

n

are written, one on each of

n

cards. The first player removes one card. Then the second player removes two cards with consecutive integers. After that the first player removes three cards with consecutive integers. Finally, the second player removes four cards with consecutive integers. What is th smallest value of

n

for which the second player can ensure that he competes both his moves?

9

1

A game on a hexagonal lattice

Olga and Sasha play a game on an infinite hexagonal grid. They take turns in placing a stone on a free hexagon of their choice. Olga starts the game. Just before the

2018

th stone is placed, a new rule comes into play. A stone may now be placed only on those free hexagons having at least two occupied neighbors. A player loses when she or he either is unable to make a move, or makes a move such that a pattern of the rhomboid shape as shown (rotated in any possible way) appears. Determine which player, if any, possesses a winning strategy.

8

1

Shooting an invisible rabbit

A graph has

N

vertices. An invisible hare sits in one of the vertices. A group of hunters tries to kill the hare. In each move all of them shoot simultaneously: each hunter shoots at a single vertex, they choose the target vertices cooperatively. If the hare was in one of the target vertices during a shoot, the hunt is finished. Otherwise the hare can stay in its vertex or jump to one of the neighboring vertices.
The hunters know an algorithm that allows them to kill the hare in at most

N!

moves. Prove that then there exists an algorithm that allows them to kill the hare in at most

2^N

moves.

7

1

Colouring a discretised torus

On a

16 \times 16

torus as shown all

512

edges are colored red or blue. A coloring is good if every vertex is an endpoint of an even number of red edges. A move consists of switching the color of each of the

4

edges of an arbitrary cell. What is the largest number of good colorings so that none of them can be converted to another by a sequence of moves?

6

1

A weird and a normal elf

Let

n

be a positive integer. Elfie the Elf travels in

\mathbb{R}^3

. She starts at the origin:

(0,0,0)

. In each turn she can teleport to any point with integer coordinates which lies at distance exactly

\sqrt{n}

from her current location. However, teleportation is a complicated procedure: Elfie starts off normal but she turns strange with her first teleportation. Next time she teleports she turns normal again, then strange again... etc. For which

n

can Elfie travel to any point with integer coordinates and be normal when she gets there?

20

1

A divisibility condition and a sum being prime

Find all the triples of positive integers

(a,b,c)

for which the number

\frac{(a+b)^4}{c}+\frac{(b+c)^4}{a}+\frac{(c+a)^4}{b}

is an integer and

a+b+c

is a prime.

19

1

An infinite set of integers with a certain property

An infinite set

B

consisting of positive integers has the following property. For each

a,b \in B

with

a>b

the number

\frac{a-b}{(a,b)}

belongs to

B

. Prove that

B

contains all positive integers. Here,

(a,b)

is the greatest common divisor of numbers

a

and

b

.

18

1

A prime dividing a large number

Let

n \ge 3

be an integer such that

4n+1

is a prime number. Prove that

4n+1

divides

n^{2n}-1

.

17

1

A new variant of an old problem on rational approximations to square roots

Prove that for any positive integers

p,q

such that

\sqrt{11}>\frac{p}{q}

, the following inequality holds:

\sqrt{11}-\frac{p}{q}>\frac{1}{2pq}.

16

1

Primes and perfect squares

Let

p

be an odd prime. Find all positive integers

n

for which

\sqrt{n^2-np}

is a positive integer.

15

1

The locus of orthocenters

Two circles in the plane do not intersect and do not lie inside each other. We choose diameters

A_1B_1

and

A_2B_2

of these circles such that the segments

A_1A_2

and

B_1B_2'

intersect. Let

A

and

B

be the midpoints of the segments

A_1A_2

and

B_1B_2

, and

C

be the intersection point of these segments. Prove that the orthocenter of the triangle

ABC

belongs to a fixed line that does not depend on the choice of diameters.

14

1

Equal segments in a circumscribed quadrilateral

A quadrilateral

ABCD

is circumscribed about a circle

\omega

. The intersection point of

\omega

and the diagonal

AC

, closest to

A

, is

E

. The point

F

is diametrally opposite to the point

E

on the circle

\omega

. The tangent to

\omega

at the point

F

intersects lines

AB

and

BC

in points

A_1

and

C_1

, and lines

AD

and

CD

in points

A_2

and

C_2

, respectively. Prove that

A_1C_1=A_2C_2

.

13

1

Of midpoints and circumcenters

The bisector of the angle

A

of a triangle

ABC

intersects

BC

in a point

D

and intersects the circumcircle of the triangle

ABC

in a point

E

. Let

K,L,M

and

N

be the midpoints of the segments

AB,BD,CD

and

AC

, respectively. Let

P

be the circumcenter of the triangle

EKL

, and

Q

be the circumcenter of the triangle

EMN

. Prove that

\angle PEQ=\angle BAC

.

12

1

Geometry with right angles and length conditions

The altitudes

BB_1

and

CC_1

of an acute triangle

ABC

intersect in point

H

. Let

B_2

and

C_2

be points on the segments

BH

and

CH

, respectively, such that

BB_2=B_1H

and

CC_2=C_1H

. The circumcircle of the triangle

B_2HC_2

intersects the circumcircle of the triangle

ABC

in points

D

and

E

. Prove that the triangle

DEH

is right-angled.

11

1

A funny twist of geometry with number theory

The points

A,B,C,D

lie, in this order, on a circle

\omega

, where

AD

is a diameter of

\omega

. Furthermore,

AB=BC=a

and

CD=c

for some relatively prime integers

a

and

c

. Show that if the diameter

d

of

\omega

is also an integer, then either

d

or

2d

is a perfect square.

5

1

Generating polynomials

A polynomial

f(x)

with real coefficients is called generating, if for each polynomial

\varphi(x)

with real coefficients there exists a positive integer

k

and polynomials

g_1(x),\dotsc,g_k(x)

with real coefficients such that

\varphi(x)=f(g_1(x))+\dotsc+f(g_k(x)).

Find all generating polynomials.

4

1

Functional equation with squares

Find all functions

f:[0, \infty) \to [0,\infty)

, such that for any positive integer

n

and and for any non-negative real numbers

x_1,x_2,\dotsc,x_n

f(x_1^2+\dotsc+x_n^2)=f(x_1)^2+\dots+f(x_n)^2.

3

1

Hard cyclic inequality with square roots

Let

a,b,c,d

be positive real numbers such that

abcd=1

. Prove the inequality

\frac{1}{\sqrt{a+2b+3c+10}}+\frac{1}{\sqrt{b+2c+3d+10}}+\frac{1}{\sqrt{c+2d+3a+10}}+\frac{1}{\sqrt{d+2a+3b+10}} \le 1.

2

1

A table with unbalanced column sums

A

100 \times 100

table is given. For each

k, 1 \le k \le 100

, the

k

-th row of the table contains the numbers

1,2,\dotsc,k

in increasing order (from left to right) but not necessarily in consecutive cells; the remaining

100-k

cells are filled with zeroes. Prove that there exist two columns such that the sum of the numbers in one of the columns is at least

19

times as large as the sum of the numbers in the other column.

1

About balanced sets of real numbers

A finite collection of positive real numbers (not necessarily distinct) is balanced if each number is less than the sum of the others. Find all

m \ge 3

such that every balanced finite collection of

m

numbers can be split into three parts with the property that the sum of the numbers in each part is less than the sum of the numbers in the two other parts.

2018 Baltic Way

Subcontests

Removing successive numbers from an interval.

A game on a hexagonal lattice

Shooting an invisible rabbit

Colouring a discretised torus

A weird and a normal elf

A divisibility condition and a sum being prime

An infinite set of integers with a certain property

A prime dividing a large number

A new variant of an old problem on rational approximations to square roots

Primes and perfect squares

The locus of orthocenters

Equal segments in a circumscribed quadrilateral

Of midpoints and circumcenters

Geometry with right angles and length conditions

A funny twist of geometry with number theory

Generating polynomials

Functional equation with squares

Hard cyclic inequality with square roots

A table with unbalanced column sums

About balanced sets of real numbers