Two lines on the plane intersect at an angle . 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 the value is a rational number.
Several points are given and for some pairs of these points the vectors are taken, and at each point the same number of vectors begin and end. Prove that the sum of all the chosen vectors is .
On the sides , and of the triangle points , and are chosen so that the segments , and intersect at one point and Prove that , and are the midpoints of the sides of the triangle .