1. The Big Apple got wormy. A worm dug a tunnel of length 101 miles inside it, and went out (starting and ending at the surface). Assuming that The Big Apple is an ideal sphere of radius 51 miles, prove that it can be cut into two congruent pieces one of which is not wormy.

2. Each vertex of a convex polyhedron is the endpoint of an even number of edges. Prove that any plane section of the polyhedron not containing a vertex is a polygon with an even number of sides.

3. Asterisks are placed in some cells of an m by n rectangular table, where m < n, so that there is at least one asterisk in each column. Prove that there exists an asterisk such that there are more asterisks in its row than in its column.

Solutions

1. Let A and B be the beginning and the end of the tunnel. Consider the set of points X such that AX + BX ≤ 101. It is an ellipsoid of rotation E with foci A and B. Each point X of the tunnel is contained in E. Indeed, since AX andXB do not exceed the lengths of the worm's routes from A to X and from X to B, respectively, we have AX + XB ≤ 101. On the other hand, the center O of the Big Apple is outside E,because AO + BO equals 51+51 = 102 > 101. Everything is clear now. Because E is convex and O is outside it, there is a plane through O that does not intersect E.