Practice library
Problems
1In how many ways can three rooks be placed on a chessboard of dimensions \(6 \times 2006\) so that no two of them attack each other? (Two rooks attack each other when they stand in the same row or in the …Open2How many three-digit numbers written using the digits \(0, 1, 2, 3, 4, 5\) are divisible by \(15\), if (a) all digits must be distinct? (b) digits may repeat?3A delegation of \(6\) people is to be chosen from a group of \(16\), consisting of \(4\) people from Serbia, \(4\) from Romania, \(4\) from Bulgaria and \(4\) from Macedonia. (a) In how many ways can this …4How many equivalence relations on a set of six elements have the property that every equivalence class contains at least two elements?5A row contains \(2016\) chairs. Each chair is to be painted either red or blue. In how many ways can this be done so that the number of neighbouring pairs of chairs painted in the same colour is even?6A park has the shape of a square with side \(1\) km. Inside it grow \(4567\) trees, each of diameter at most \(50\) cm, and each tree lies entirely within the park. Prove that the park contains a \(10\) …7Eight players took part in a chess tournament, and each of them played exactly one game against every other participant. A win earns \(1\) point, a loss \(0\) points, and a draw \(0.5\) points for each …8A mosquito sits on the lower left cell of a rectangular board of format \(2003 \times 2004\). It travels above the board in the following manner: taking off from the cell it occupies, it flies over \(99\) …9Into a box, \(k\) smaller boxes are placed. Then \(k\) still smaller boxes are placed into some of the smaller boxes, each, and this procedure is repeated several times. If, at the end, \(m\) of all these …10A car leaves town \(A\) and drives along a straight road at constant speed. Every \(15\) minutes it makes a turn of \(90\) degrees, to the left or to the right. Prove that the car can be back in \(A\) …11In how many ways can \(11\) birds be placed into \(3\) identical cages so that every cage contains at least three birds?12Ten teams took part in a volleyball tournament, and each team played exactly one match against each of the other nine. When the tournament ended, the first team had \(x_{1}\) wins and \(y_{1}\) losses, …13A snake starts in the upper-left cell of a \(2 \times n\) board, where \(n\) is a natural number. From one cell it may move to another whenever the two cells share an edge, but it may never visit a cell …14Aca and Branko play the following game on a \(2023 \times 2024\) board. First Aca chooses a square of the board and places a queen on it. Then the players move the queen alternately, following the rules …
Showing 14 of 651 - problem statements are free for everyone.