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Part of 2017 Czech-Polish-Slovak Match

Problems(2)

Pepa & Geoff, game strategy with contiguous subsequences

Source: Czech-Polish-Slovak Match 2017 day 1 P3

9/28/2017
Let k{k} be a fi xed positive integer. A finite sequence of integers x1,x2,...,xn{x_1,x_2, ..., x_n} is written on a blackboard. Pepa and Geoff are playing a game that proceeds in rounds as follows. - In each round, Pepa first partitions the sequence that is currently on the blackboard into two or more contiguous subsequences (that is, consisting of numbers appearing consecutively). However, if the number of these subsequences is larger than 2{2}, then the sum of numbers in each of them has to be divisible by k{k}. - Then Geoff selects one of the subsequences that Pepa has formed and wipes all the other subsequences from the blackboard. The game fi nishes once there is only one number left on the board. Prove that Pepa may choose his moves so that independently of the moves of Geoff, the game fi nishes after at most 3k{3k} rounds.
(Poland)
winning positionsgame strategySequencedivisiblenumber theory
f(x) - f(x+ y) = f ( x / y) f(x + y) with x,y >0

Source: Czech-Polish-Slovak Match 2017 day 2 P3

9/28/2017
Find all functions f:(0,+∞)→R{f : (0, +\infty) \rightarrow R} satisfying f(x)−f(x+y)=f(xy)f(x+y)f(x) - f(x+ y) = f \left( \frac{x}{y}\right) f(x + y) for all x,y>0x, y > 0.
(Austria)
functional equationpositiveFind all functionsalgebra