2019 Kosovo Team Selection Test

Part of Kosovo Team Selection Test

Subcontests

(3)

1

Kosovo TST 2019 P5

a,b,c,d

are fixed positive real numbers. Find the maximum value of the function

f: \mathbb{R^{+}}_{0} \rightarrow \mathbb{R}

f(x)=\frac{a+bx}{b+cx}+\frac{b+cx}{c+dx}+\frac{c+dx}{d+ax}+\frac{d+ax}{a+bx}, x \geq 0

1

Kosovo TST 2019 P2

Determine all functions

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

such that for every

x,y \in \mathbb{R}

f(x^{4}-y^{4})+4f(xy)^{2}=f(x^{4}+y^{4})

1

Kosovo TST 2019 P1

There are 2019 cards in a box. Each card has a number written on one of its sides and a letter on the other side. Amy and Ben play the following game: in the beginning Amy takes all the cards, places them on a line and then she flips as many cards as she wishes. Each time Ben touches a card he has to flip it and its neighboring cards. Ben is allowed to have as many as 2019 touches. Ben wins if all the cards are on the numbers' side, otherwise Amy wins. Determine who has a winning strategy.