Problems

The distance from school to the monument in the town centre is \(4.2\) km and the distance from Anna’s house to school is \(0.7\) km. Given that the distance from Anna’s house to the monument is an integer number of kilometres, what is this distance?

Tom and his grandma live on the same side of a straight river. Tom wants to visit his grandma, but also wants to stop by the river and fill his bottle with water. What is the shortest path that starts at his house, touches the river and ends at his grandma’s house?

A point \(P\) is somewhere inside the triangle \(\triangle ABC\). Show that \(AP + BP < AC + BC\).

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 for any three points on the plane \(A,B\) and \(C\), \(AB \ge |BC - AC|\).

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?