6th Iberoamerican 1991


A1. The number 1 or the number -1 is assigned to each vertex of a cube. Then each face is given the product of its four vertices. What are the possible totals for the resulting 14 numbers?
A2. Two perpendicular lines divide a square into four parts, three of which have area 1. Show that the fourth part also has area 1.
A3. f is a function defined on all reals in the interval [0, 1] and satisfies f(0) = 0, f(x/3) = f(x)/2, f(1 - x) = 1 - f(x). Find f(18/1991).
B1. Find a number N with five digits, all different and none zero, which equals the sum of all distinct three digit numbers whose digits are all different and are all digits of N.
B2. Let p(m, n) be the polynomial 2m² - 6mn + 5n². The range of p is the set of all integers k such that k = p(m, n) for some integers m, n. Find which members of {1, 2, ... , 100} are in the range of p. Show that if h and k are in the range of p, then so is hk.
B3. Given three non-collinear points M, N, H show how to construct a triangle which has H as orthocenter and M and N as the midpoints of two sides.

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