Two people take turns drawing noughts and crosses on a \(9 \times 9\) grid. The first player uses crosses and the second player uses noughts. After they finish, the number of rows and columns where there are more crosses than noughts are counted, and these are the points which the first player receives. The number of rows and columns where there are more noughts than crosses are the second player’s points. The player who has the most points is the winner. Who wins, if the right strategy is used?
A rook is on the a1 square of a chessboard. Consider the game with two players where: in one move a player can move the rook by any number of squares to the left, right or up. The winner is the player who places the rook on the square h8. Who would win, if the right strategy is used?
Prove that: \[a_1 a_2 a_3 \cdots a_{n-1}a_n \times 10^3 \equiv a_{n-1} a_n \times 10^3 \pmod4,\] where \(n\) is a natural number and \(a_i\) for \(i=1,2,\ldots, n\) are the digits of some number.
Solve the equation \(3x + 5y = 7\) in integers.
Determine all integer solutions of the equation \(3x - 12y = 7\).
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 \(ab \equiv 1 \pmod p\).
How many ways can Susan choose 4 colours from 7 different ones?