101stKürschák 2001


1. Given any 3n-1 points in the plane, no three collinear, show that it is possible to find 2n whose convex hull is not a triangle.
2. k > 2 is an integer and n > kC3 (where aCb is the usual binomial coefficient a!/(b! (a-b)!) ). Show that given 3n distinct real numbers a_i, b_i, c_i (where i = 1, 2, ... , n), there must be at least k+1 distinct numbers in the set {a_i + b_i, b_i + c_i, c_i + a_i \| i = 1, 2, ... , n}. Show that the statement is not always true for n = kC3.
3. The vertices of the triangle ABC are lattice points and there is no smaller triangle similar to ABC with its vertices at lattice points. Show that the circumcenter of ABC is not a lattice point.

Thanks to Shrenik Shah for finding the problems in English. The original problems are in Hungarian.