Subcontests
(4)the broken line A_1A_2...A_n does not intersect itself
Suppose there are n points in a plane no three of which are collinear with the property that if we label these points as A1,A2,…,An in any way whatsoever, the broken line A1A2…An does not intersect itself. Find the maximum value of n.
Dinu Serbanescu, Romania Why MN is parallel to BC?
Let D, E, F be the midpoints of the arcs BC, CA, AB on the circumcircle of a triangle ABC not containing the points A, B, C, respectively. Let the line DE meets BC and CA at G and H, and let M be the midpoint of the segment GH. Let the line FD meet BC and AB at K and J, and let N be the midpoint of the segment KJ.
a) Find the angles of triangle DMN;
b) Prove that if P is the point of intersection of the lines AD and EF, then the circumcenter of triangle DMN lies on the circumcircle of triangle PMN.