These days (26.-27.3.2015), another national round of Slovak Olympiad in Informatics took place. Just short problem statements for now, I'll write it in more detail and with solutions later:

1.

There's a hazelnut chocolate of size N × M, which can be viewed as a 2D grid. In each cell of the grid, there's a given number of nuts. Two players play a game, alternating turns: each player has to cut off either the bottom row or the rightmost column of the remaining part of the chocolate in her turn (and do with it guess what). The cut off row/column has to contain an even number of nuts. The player that can't make a move loses. Determine the winner if both play optimally.

2.

You have two numbers N, K (), one input file with N - K distinct numbers between 1 and N and 2K + 2 empty files. Find the missing K numbers.

Your program has a very limited amount of memory (the files' content isn't stored in that memory), just 10 kB for K ≤ 100; its memory complexity can't depend on N. Primarily, you need to minimise the worst-case number of reads+writes to files; only then are you minimising the time and memory complexity of your algorithm.

Files work like queues and can be erased in small constant time.

3.

a very long introductory text and some heavily theoretical magic with simulating boolean conditions using other conditions and proving that it's always possible/impossible

4.

There are N numbers up to 10⁹ and another number R, also up to 10⁹. Find up to 4 numbers (one number can't be used multiple times) among these N that sum up to R or decide that they don't exist. N ≤ 4000.

5.

There's a straight river of constant width, N points on one side of it and M on the other side. You need to connect some of them with (obviously straight) lines that have the minimum total length in such a way that each of these N + M points has at least one line connecting it with another point on the other side of the river. N, M ≤ 20000.

The first 3 problems are theoretical (points for explaining your algorithm), the other 2 practical (typical contest style, partial scoring); the max. time limit was 10s in both.