MathDB

1

Integer polynomial with three fixed values

P(x)

with integers coefficients satisfying

P(-1)=-4,\ P(-3)=-40,\text{ and } P(-5)=-156.

What is the largest possible number of integers

x

satisfying

P(P(x))=x^2?

19

1

y^2+z^2+1 cannot be a power of 7

Prove that the equation

7^x=1+y^2+z^2

has no solutions over positive integers.

18

1

At least one of b^2+b+1 and c^2+c+1 is composite.

Let

a,b

, and

c

be odd positive integers such that

a

is not a perfect square and

a^2+a+1 = 3(b^2+b+1)(c^2+c+1).

Prove that at least one of the numbers

b^2+b+1

and

c^2+c+1

is composite.

17

1

Congruence equation modulo p

Let

p

be an odd prime. Show that for every integer

c

, there exists an integer

a

such that

a^{\frac{p+1}{2}} + (a+c)^{\frac{p+1}{2}} \equiv c\pmod p.

16

1

Number of solutions is always a perfect square

For a positive integer

N

, let

f(N)

be the number of ordered pairs of positive integers

(a,b)

such that the number

\frac{ab}{a+b}

is a divisor of

N

. Prove that

f(N)

is always a perfect square.

15

1

Polygons which closest vertex is not neighbor

Let

n \geq 4

, and consider a (not necessarily convex) polygon P_1P_2\hdots P_n in the plane. Suppose that, for each

P_k

, there is a unique vertex

Q_k\ne P_k

among P_1,\hdots, P_n that lies closest to it. The polygon is then said to be hostile if

Q_k\ne P_{k\pm 1}

for all

k

(where

P_0 = P_n

,

P_{n+1} = P_1

).
(a) Prove that no hostile polygon is convex. (b) Find all

n \geq 4

for which there exists a hostile

n

-gon.

14

1

Medians of right triangles form bisecting lines

Let

ABC

be a triangle with

\angle ABC = 90^{\circ}

, and let

H

be the foot of the altitude from

B

. The points

M

and

N

are the midpoints of the segments

AH

and

CH

, respectively. Let

P

and

Q

be the second points of intersection of the circumcircle of the triangle

ABC

with the lines

BM

and

BN

, respectively. The segments

AQ

and

CP

intersect at the point

R

. Prove that the line

BR

passes through the midpoint of the segment

MN

.

13

1

Perpendicular diagonals in a hexagon

Let

ABCDEF

be a convex hexagon in which

AB=AF

,

BC=CD

,

DE=EF

and

\angle ABC = \angle EFA = 90^{\circ}

. Prove that

AD\perp CE

.

12

1

Floating orthocenter forms an if and only if condition

Let

ABC

be a triangle and

H

its orthocenter. Let

D

be a point lying on the segment

AC

and let

E

be the point on the line

BC

such that

BC\perp DE

. Prove that

EH\perp BD

if and only if

BD

bisects

AE

.

11

1

Intersection of two circles in an isosceles triangle

Let

ABC

be a triangle with

AB = AC

. Let

M

be the midpoint of

BC

. Let the circles with diameters

AC

and

BM

intersect at points

M

and

P

. Let

MP

intersect

AB

at

Q

. Let

R

be a point on

AP

such that

QR \parallel BP

. Prove that

CP

bisects

\angle RCB

.

10

1

Drawing discs on a 2019-points configuration

There are

2019

points given in the plane. A child wants to draw

k

(closed) discs in such a manner, that for any two distinct points there exists a disc that contains exactly one of these two points. What is the minimal

k

, such that for any initial configuration of points it is possible to draw

k

discs with the above property?

9

1

Count all nonincreasing functions with no fixed points

For a positive integer

n

, consider all nonincreasing functions f : \{1,\hdots,n\}\to\{1,\hdots,n\}. Some of them have a fixed point (i.e. a

c

such that

f(c) = c

), some do not. Determine the difference between the sizes of the two sets of functions.
Remark. A function

f

is nonincreasing if

f(x) \geq f(y)

holds for all

x \leq y

8

1

62 cops chase a visible robber on a graph with 2019 vertices

There are

2019

cities in the country of Balticwayland. Some pairs of cities are connected by non-intersecting bidirectional roads, each road connecting exactly 2 cities. It is known that for every pair of cities

A

and

B

it is possible to drive from

A

to

B

using at most

2

roads. There are

62

cops trying to catch a robber. The cops and robber all know each others’ locations at all times. Each night, the robber can choose to stay in her current city or move to a neighbouring city via a direct road. Each day, each cop has the same choice of staying or moving, and they coordinate their actions. The robber is caught if she is in the same city as a cop at any time. Prove that the cops can always catch the robber

7

1

Partition of {2,3,...,k} into two product free sets

Find the smallest integer

k \geq 2

such that for every partition of the set \{2, 3,\hdots, k\} into two parts, at least one of these parts contains (not necessarily distinct) numbers

a

,

b

and

c

with

ab = c

.

6

1

A game with four algebraic expressions

Alice and Bob play the following game. They write the expressions

x + y

,

x - y

,

x^2+xy+y^2

and

x^2-xy+y^2

each on a separate card. The four cards are shuffled and placed face down on a table. One of the cards is turned over, revealing the expression written on it, after which Alice chooses any two of the four cards, and gives the other two to Bob. All cards are then revealed. Now Alice picks one of the variables

x

and

y

, assigns a real value to it, and tells Bob what value she assigned and to which variable. Then Bob assigns a real value to the other variable.
Finally, they both evaluate the product of the expressions on their two cards. Whoever gets the larger result, wins. Which player, if any, has a winning strategy?

5

1

Moves on the blackboard

The

2m

numbers 1\cdot 2, 2\cdot 3, 3\cdot 4,\hdots,2m(2m+1) are written on a blackboard, where

m\geq 2

is an integer. A move consists of choosing three numbers

a, b, c

, erasing them from the board and writing the single number

\frac{abc}{ab+bc+ca}

After

m-1

such moves, only two numbers will remain on the blackboard. Supposing one of these is

\tfrac{4}{3}

, show that the other is larger than

4

.

4

1

Find all possible values of x_1x_2 + x_2x_3 + ... + x_{k-1}x_k

Determine all integers

n

for which there exist an integer

k\geq 2

and positive integers x_1,x_2,\hdots,x_k so that x_1x_2+x_2x_3+\hdots+x_{k-1}x_k=n\text{ and } x_1+x_2+\hdots+x_k=2019.

3

1

Boring Functional Equation

Find all functions

f:\mathbb{R}\to\mathbb{R}

such that

f(xf(y)-y^2)=(y+1)f(x-y)

holds for all

x,y\in\mathbb{R}

.

2

1

Fibonacci Diophantine

Let

(F_n)

be the sequence defined recursively by

F_1=F_2=1

and

F_{n+1}=F_n+F_{n-1}

for

n\geq 2

. Find all pairs of positive integers

(x,y)

such that

5F_x-3F_y=1.

1

Asymmetric Inequality

For all non-negative real numbers

x,y,z

with

x \geq y

, prove the inequality

\frac{x^3-y^3+z^3+1}{6}\geq (x-y)\sqrt{xyz}.

2019 Baltic Way

Subcontests

Integer polynomial with three fixed values

y^2+z^2+1 cannot be a power of 7

At least one of b^2+b+1 and c^2+c+1 is composite.

Congruence equation modulo p

Number of solutions is always a perfect square

Polygons which closest vertex is not neighbor

Medians of right triangles form bisecting lines

Perpendicular diagonals in a hexagon

Floating orthocenter forms an if and only if condition

Intersection of two circles in an isosceles triangle

Drawing discs on a 2019-points configuration

Count all nonincreasing functions with no fixed points

62 cops chase a visible robber on a graph with 2019 vertices

Partition of {2,3,...,k} into two product free sets

A game with four algebraic expressions

Moves on the blackboard

Find all possible values of x_1x_2 + x_2x_3 + ... + x_{k-1}x_k

Boring Functional Equation

Fibonacci Diophantine

Asymmetric Inequality