Problems

Two lines on the plane intersect at an angle $\alpha$. On one of them there is a flea. Every second it jumps from one line to the other (the point of intersection is considered to belong to both straight lines). It is known that the length of each of her jumps is 1 and that she never returns to the place where she was a second ago. After some time, the flea returned to its original point. Prove that for the angle $\alpha$ the value $\alpha /\pi$ is a rational number.

In a square with side length 1 there is a broken line, which does not self-intersect, whose length is no less than 200. Prove that there is a straight line parallel to one of the sides of the square that intersects the broken line at a point no less than 101 units along the line.

In a board, 20 pins are placed (see the picture). The distance between any adjacent pins is 1 inch. Pull a string of length 19 inches from the first pin to the second one, so that it goes through all the pins.

Is it possible to draw this picture (see the figure), without taking your pencil off the paper and going along each line only once?

In the middle of an empty pool there is a square platform of $50\times 50$ cm, split into cells of $10\times 10$ cm. Sunny builds towers of $10\times 10\times 10$cm cubes on the platform cells. After that his friend Margo turns on the water and counts how many towers are still above the water level. They call each visible tower an island.

For example, let’s consider the case when the heights of the towers are as given in the table on the right. Then at the water level of $5$ cm there is $1$ island, at the water level of $15$ cm there are two islands (if the islands have a common corner or don’t intersect at all, they are considered separate islands), and at the water level of $25$ cm, all the towers are covered with water and there are $0$ islands.

Find out how Sunny should build his towers to get the following numbers of islands corresponding to the level of water in the pool: $$\begin{array}{cccccc}\mathit{\text{Water level (cm)}}& 5& 15& 25& 35& 45\\ \mathit{\text{Number of islands}}& 2& 5& 2& 5& 0\end{array}$$

In the solution, write down how many cubes are there composing a tower in each cell as it is done in the example.