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?

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?

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

Two villages lie on the opposite sides of a river whose banks are straight lines. A bridge is to be built over the river perpendicular to the banks. Where should the bridge be built so that the path from one village to the other is as short as possible?

Quadrilateral $ABCD$ is situated completely inside a quadrilateral $EFGH$. Prove that the perimeter of $ABCD$ is smaller than the perimeter of $EFGH$.

There are $n$ mines and $n$ cities scattered across the land, it is known that no three objects (mines, or cities) belong to one line. Every mine has to have a rail connection to exactly one city. Railways have to be straight and cannot cross other railways. Is it always possible?

Prove the triangle inequality: in any triangle $ABC$ the side $AB<AC+BC$.

In certain kingdom there are a lot of cities, it is known that all the distances between the cities are distinct. One morning one plane flew out of each city to the nearest city. Could it happen that in one city landed more than $5$ planes?