33rd Spanish 1997


A1. An arithmetic progression has 100 terms. The sum of the terms is -1, and the sum of the even-numbered terms is 1. Find the sum of the squares of the terms.
A2. X is the set of 16 points shown. What is the largest number of elements of X that we can choose so that no three of the chosen points form an isosceles triangle?
A3. Let S be the set of all parabolas y = x² + px + q whose graphs cut the coordinate axes in three distinct points. Let C(p,q) be the circle through the three points. Show that all circles C(p,q) have a common point.
B1. Given a prime p, find all integers k such that √(k²-kp) is integral.
B2. Q is a convex quadrilateral with area 1. Show that the sum of the sides and diagonals is at least 2(2+√2).
B3. A car wishes to make a circuit of a circular road. There are some tanks along the road that contain between them just enough gasoline for the car to make the trip. The car has a tank large enough to hold all the gasoline for a complete circuit, but the tank is initially empty. Show that irrespective of the number of tanks, their positions, and the amount each contains, it is possible to find a starting point on the road which will allow the car to make a complete circuit (refueling when it reaches a tank).

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