Several circles, whose total length of circumferences is 10, are placed inside a square of side 1. Prove that there will always be some straight line that crosses at least four of the circles.
On a circle of radius 1, the point \(O\) is marked and from this point, to the right, a notch is marked using a compass of radius \(l\). From the obtained notch \(O_1\), a new notch is marked, in the same direction with the same radius and this is process is repeated 1968 times. After this, the circle is cut at all 1968 notches, and we get 1968 arcs. How many different lengths of arcs can this result in?
In a regular shape with 25 vertices, all the diagonals are drawn.
Prove that there are no nine diagonals passing through one interior point of the shape.