9th Iberoamerican 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 + BC. What is the maximum possible area of ABCD in terms of \|AB\| and \|CD\|?
A3. There is a bulb in each cell of an n x n board. Initially all the bulbs are off. If a bulb is touched, that bulb and all the bulbs in the same row and column change state (those that are on, turn off, and those that are off, turn on). Show that it is possible by touching m bulbs to turn all the bulbs on. What is the minimum possible value of m?
B1. ABC is an acute-angled triangle. P is a point inside its circumcircle. The rays AP, BP, CP intersect the circle again at D, E, F. Find P so that DEF is equilateral.
B2. n and r are positive integers. Find the smallest k for which we can construct r subsets A₁, A₂, ... , A_r of {0, 1, 2, ... , n-1} each with k elements such that each integer 0 ≤ m < n can be written as a sum of one element from each of the r subsets.
B3. Show that given any integer 0 < n ≤ 2^1000000 we can find at set S of at most 1100000 positive integers such that S includes 1 and n and every element of S except 1 is a sum of two (possibly equal) smaller elements of S.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.