OBM 1983

Show that there are only finitely many solutions to $1/a + 1/b + 1/c = 1/1983$ in positive integers.

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.

Show that $1 + 1/2 + 1/3 + ... + 1/n$ is not an integer for $n > 1$ .

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.

Show that $1 \le n^{1/n} \le 2$ for all positive integers $n$ . Find the smallest $k$ such that $1 \le n ^{1/n} \le k$ for all positive integers $n$ .

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 $14$ .