Problems
Determine all integer solutions of the equation \(3x - 12y = 7\).
A pawn stands on one of the squares of an endless in both directions chequered strip of paper. It can be shifted by \(m\) squares to the right or by \(n\) squares to the left. For which \(m\) and \(n\) can it move to the next cell to the right?
Solve the equation with integers \(x^2 + y^2 = 4z - 1\).
a) Two students need to be chosen to participate in a mathematical Olympiad from a class of 30 students. In how many ways can this be done?
b) In how many ways can a team of three students in the same class be chosen?
A person has 10 friends and within a few days invites some of them to visit so that his guests never repeat (on some of the days he may not invite anyone). How many days can he do this for?
Prove that out of \(n\) objects an even number of objects can be chosen in \(2^{n-1}\) ways.
Prove that every number \(a\) in Pascal’s triangle is equal to
a) the sum of the numbers of the previous right diagonal, starting from the leftmost number up until the one to the right above the number \(a\).
b) the sum of the numbers of the previous left diagonal, starting from the leftmost number to the one to left of the number which is above \(a\).
How many necklaces can be made from five identical red beads and two identical blue beads?
How many ways can you build a closed line whose vertices are the vertices of a regular hexagon (the line can be self-intersecting)?