Problems

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 triangle $ABC$, the angle $A$ is equal to ${40}^{\circ }$. The triangle is randomly thrown onto a table. Find the probability that the vertex $A$ lies east of the other two vertices.

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.