9th Iberoamerican Mathematical Olympiad Problems 1994

A1.  Show that there is a number 1 < b < 1993 such that if 1994 is written in base b then all its digits are the same. Show that there is no number 1 < b < 1992 such that if 1993 is written in base b then all its digits are the same. A2.  ABCD is a cyclic quadrilateral. A circle whose center is on the side AB touches the other three sides. Show that AB = AD