11th Iberoamerican 1996

11th Iberoamerican 1996 problems


A1. Find the smallest positive integer n so that a cube with side n can be divided into 1996 cubes each with side a positive integer.
A2. M is the midpoint of the median AD of the triangle ABC. The ray BM meets AC at N. Show that AB is tangent to the circumcircle of NBC iff BM/MN = (BC/BN)².
A3. n = k² - k + 1, where k is a prime plus one. Show that we can color some squares of an n x n board black so that each row and column has exactly k black squares, but there is no rectangle with sides parallel to the sides of the board which has its four corner squares black.
B1. n > 2 is an integer. Consider the pairs (a, b) of relatively prime positive integers, such that a < b ≤ n and a + b > n. Show that the sum of 1/ab taken over all such pairs is 1/2.
B2. An equilateral triangle of side n is divided into n² equilateral triangles of side 1 by lines parallel to the sides. Initially, all the sides of all the small triangles are painted blue. Three coins A, B, C are placed at vertices of the small triangles. Each coin in turn is moved a distance 1 along a blue side to an adjacent vertex. The side it moves along is painted red, so once a coin has moved along a side, the side cannot be used again. More than one coin is allowed to occupy the same vertex. The coins are moved repeatedly in the order A, B, C, A, B, C, ... . Show that it is possible to paint all the sides red in this way.
B3. A₁, A₂, ... , A_n are points in the plane. A non-zero real number k_i is assigned to each point, so that the square of the distance between A_i and A_j (for i ≠ j) is k_i + k_j. Show that n is at most 4 and that if n = 4, then 1/k₁ + 1/k₂ + 1/k₃ + 1/k₄ = 0.

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