8th Iberoamerican 1993

8th Iberoamerican 1993 problems


A1. A palindrome is a positive integers which is unchanged if you reverse the order of its digits. For example, 23432. If all palindromes are written in increasing order, what possible prime values can the difference between successive palindromes take?
A2. Show that any convex polygon of area 1 is contained in some parallelogram of area 2.
A3. Find all functions f on the positive integers with positive integer values such that (1) if x < y, then f(x) < f(y), and (2) f(y f(x)) = x²f(xy).
B1. ABC is an equilateral triangle. D is on the side AB and E is on the side AC such that DE touches the incircle. Show that AD/DB + AE/EC = 1.
B2. If P and Q are two points in the plane, let m(PQ) be the perpendicular bisector of PQ. S is a finite set of n > 1 points such that: (1) if P and Q belong to S, then some point of m(PQ) belongs to S, (2) if PQ, P'Q', P"Q" are three distinct segments, whose endpoints are all in S, then if there is a point in all of m(PQ), m(P'Q'), m(P"Q") it does not belong to S. What are the possible values of n?
B3. We say that two non-negative integers are related if their sum uses only the digits 0 and 1. For example 22 and 79 are related. Let A and B be two infinite sets of non-negative integers such that: (1) if a ∈ A and b ∈ B, then a and b are related, (2) if c is related to every member of A, then it belongs to B, (3) if c is related to every member of B, then it belongs to A. Show that in one of the sets A, B we can find an infinite number of pairs of consecutive numbers.

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