Problems
One side of a triangle has length \(1\), the second has length \(4\) and the third one has integer length. What is the side length of the third side?
The distance between London and Warsaw equals \(1450\) km, between Warsaw and Kyiv is \(680\) km. The distance from London to New Delhi, is \(6700\) km and the distance from Kyiv to New Delhi is \(4570\) km. What is the distance from London to Kyiv?
Show that if all sides of a triangle have integer lengths and one of them is equal to \(1\), then the other two have lengths equal to each other.
A billiard ball lies on a table in the shape of an acute angle. How should you hit the ball so that it returns to its starting location after hitting each of the two banks once? Is it always possible to do so?
(When the ball hits the bank, it bounces. The way it bounces is determined by the shortest path rule – if it begins at some point \(D\) and ends at some point \(D'\) after bouncing, the path it takes is the shortest possible path that includes the bounce.)
Can you cover a \(10 \times 10\) board using only \(T\)-shaped tetraminoes?
A broken calculator can only do several operations: multiply by 2, divide by 2, multiply by 3, divide by 3, multiply by 5, and divide by 5. Using this calculator any number of times, could you start with the number 12 and end up with 49?
Can you cover a \(10 \times 10\) square with \(1 \times 4\) rectangles?
Two opposite corners were removed from an \(8 \times 8\) chessboard. Can you cover this chessboard with \(1 \times 2\) rectangular blocks?
The numbers 1 through 12 are written on a board. You can erase any two of these numbers (call them \(a\) and \(b\)) and replace them with the number \(a+b-1\). After 11 such operations, there will be just one number left. What could this number be?
If a magician puts 1 dove into his hat, he pulls out 2 rabbits and 2 flowers from it. If the magician puts 1 rabbit in, he pulls out 2 flowers and 2 doves. If he puts 1 flower in, he pulls out 1 rabbit and 3 doves. The magician starts with 1 rabbit. Could he end up with the same number of rabbits, doves, and flowers after performing his hat trick several times?