KURSCHAK 1970

What is the largest possible number of acute angles in an $n$ -gon which is not selfintersecting (no two non-adjacent edges interesect)?

A valid lottery ticket is formed by choosing $5$ distinct numbers from $1, 2,3,..., 90$ . What is the probability that the winning ticket contains at least two consecutive numbers?

n points are taken in the plane, no three collinear. Some of the line segments between the points are painted red and some are painted blue, so that between any two points there is a unique path along colored edges. Show that the uncolored edges can be painted (each edge either red or blue) so that all triangles have an odd number of red sides.