The kingdom of Triangland is an equilateral triangle of side \(10\) km. There are \(5\) cities in this kingdom. Show that some two of them are closer than \(5\) km apart.
Margaret marked three points with integer coordinates on a number line with a red crayon. Meanwhile Angelina marked the midpoint of each pair of red points with a blue crayon. Prove that at least one of the blue points has an integer coordinate.
Margaret and Angelina coloured points in the second dimension. Now Margaret marked five points with both integer coordinates on a plane with a red crayon, while Angelina marked the midpoint for each pair of red points with a blue crayon. Prove that at least one of the blue points has both integer coordinates.
Anna has a garden of square shape with side \(4\) m. After playing with her dog in the garden she left \(5\) dog toys on the lawn. Show that some two of them are closer than \(3\) m apart.