MathDB

[url=https://artofproblemsolving.com/community/c678145]Czech-Polish-Slovak Match 2018 Austria, 24 - 27 June 2018
[url=http://artofproblemsolving.com/community/c6h1667029p10595005]Problem 1. Determine all functions

f : \mathbb R \to \mathbb R

such that for all real numbers

x

and

y

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

Proposed by Walther Janous, Austria
[url=http://artofproblemsolving.com/community/c6h1667030p10595011]Problem 2. Let

ABC

be an acute scalene triangle. Let

D

and

E

be points on the sides

AB

and

AC

, respectively, such that

BD=CE

. Denote by

O_1

and

O_2

the circumcentres of the triangles

ABE

and

ACD

, respectively. Prove that the circumcircles of the triangles

ABC, ADE

, and

AO_1O_2

have a common point different from

A

.
Proposed by Patrik Bak, Slovakia
[url=http://artofproblemsolving.com/community/c6h1667031p10595016]Problem 3. There are

2018

players sitting around a round table. At the beginning of the game we arbitrarily deal all the cards from a deck of

K

cards to the players (some players may receive no cards). In each turn we choose a player who draws one card from each of the two neighbors. It is only allowed to choose a player whose each neighbor holds a nonzero number of cards. The game terminates when there is no such player. Determine the largest possible value of

K

such that, no matter how we deal the cards and how we choose the players, the game always terminates after a finite number of turns.
Proposed by Peter Novotný, Slovakia
[url=http://artofproblemsolving.com/community/c6h1667033p10595021]Problem 4. Let

ABC

be an acute triangle with the perimeter of

2s

. We are given three pairwise disjoint circles with pairwise disjoint interiors with the centers

A, B

, and

C

, respectively. Prove that there exists a circle with the radius of

s

which contains all the three circles.
Proposed by Josef Tkadlec, Czechia
[url=http://artofproblemsolving.com/community/c6h1667034p10595023]Problem 5. In a

2 \times 3

rectangle there is a polyline of length

36

, which can have self-intersections. Show that there exists a line parallel to two sides of the rectangle, which intersects the other two sides in their interior points and intersects the polyline in fewer than

10

points.
Proposed by Josef Tkadlec, Czechia and Vojtech Bálint, Slovakia
[url=http://artofproblemsolving.com/community/c6h1667036p10595032]Problem 6. We say that a positive integer

n

is fantastic if there exist positive rational numbers

a

and

b

such that

n = a + \frac 1a + b + \frac 1b.

(a) Prove that there exist infinitely many prime numbers

p

such that no multiple of

p

is fantastic. (b) Prove that there exist infinitely many prime numbers

p

such that some multiple of

p

is fantastic.
Proposed by Walther Janous, Austria

Problem Statement