7
Problems(2)
The Magician Fails
Source: 239 2015 S P7
5/14/2020
Two magicians are about to show the next trick. A circle is drawn on the board with one semicircle marked. Viewers mark 100 points on this circle, then the first magician erases one of them. After this, the second one for the first time looks at the drawing and determines from the remaining 99 points whether the erased point was lying on the marked semicircle. Prove that such a trick will not always succeed.
Magiciancombinatorics
polyline which edges connect the midpoints of two adjacent edges of the previous
Source: 239 2015 J7
5/17/2020
There is a closed polyline with edges on the plane. We build a new polyline which edges connect the midpoints of two adjacent edges of the previous polyline. Then we erase previous polyline and start over and over. Also we know that each polyline satisfy that all vertices are different and not all of them are collinear. For which we can get a polyline that is a сonvex polygon?
combinatorics