Consider a chessboard of size (number of rows \(\times\) number of columns): a) \(9\times 10\); b) \(10\times 12\); c) \(9\times 11\). Two people are playing a game where: in one turn a player is allowed to cross out any row or column as long as there it contains at least one square that is not crossed out. The loser is the player who cannot make a move. Which player will win?
Two people take turns placing bishops on a chessboard such that the bishops cannot attack each other. Here, the colour of the bishops does not matter. (Note: bishops move and attack diagonally.) Which player wins the game, if the right strategy is used?
Solve the equation \(3x + 5y = 7\) in integers.
Determine all the integer solutions for the equation \(21x + 48y = 6\).
Solve the equations \(x^2 = 14 + y^2\) in integers.
Solve the equation with integers \(x^2 + y^2 = 4z - 1\).
Let \(p\) be a prime number, and \(a\) an integer number not divisible by \(p\). Prove that there is a positive integer \(b\) such that \(ab \equiv 1 \pmod p\).
30 people vote on five proposals. In how many ways can the votes be distributed if everyone votes only for one proposal and only the number of votes cast for each proposal is taken into account?
a) they have 10 vertices, the degree of each of which is equal to 9?
b) they have 8 vertices, the degree of each of which is equal to 3?
c) are they connected, without cycles and contain 6 edges?
On the plane 100 circles are given, which make up a connected figure (that is, not falling apart into pieces). Prove that this figure can be drawn without taking the pencil off of the paper and going over any line twice.