5th Brasil 1983


1. Show that there are only finitely many solutions to 1/a + 1/b + 1/c = 1/1983 in positive integers.
2. An equilateral triangle ABC has side a. A square is constructed on the outside of each side of the triangle. A right regular pyramid with sloping side a is placed on each square. These pyramids are rotated about the sides of the triangle so that the apex of each pyramid comes to a common point above the triangle. Show that when this has been done the other vertices of the bases of the pyramids (apart from the vertices of the triangle) form a regular hexagon.
3. Show that 1 + 1/2 + 1/3 + ... + 1/n is not an integer for n > 1.
4. Show that it is possible to color each point of a circle red or blue so that no right-angled triangle inscribed in the circle has its vertices all the same color.
5. Show that 1 ≤ n^1/n ≤ 2 for all positive integers n. Find the smallest k such that 1 ≤ n^1/n ≤ k for all positive integers n.
6. Show that the maximum number of spheres of radius 1 that can be placed touching a fixed sphere of radius 1 so that no pair of spheres has an interior point in common is between 12 and 14.

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