32nd Eötvös Competition Problems 1928

1.  Show that for any real x, at least one of x, 2x, 3x, ... , (n-1)x differs from an integer by no more than 1/n. 2.  The numbers 1, 2, ... , n are arranged around a circle so that the difference between any two adjacent numbers does not exceed 2. Show that this can be done in only one way (treating rotations and reflections of an arrangement as the same