From Peter Winkler's 'Mathematical puzzles', taken from an All USSR Mathematical Competition, 1976:
50 accurate watches lie on a table. Prove that there exists a moment in time when the sum of the distances from the center of the table to the ends of the minute hands is more than the sum of the distances from the center of the table to the centers of the watches
There is an answer in the book, but it is rather more complicated than my attempt below:
Assume there are n watches to the right of the center of the table and m watches to the left. Call L the length of the hands. Call D the sum of the distance to the center of all the watches. Then at 45 past, the distance from the center to the hands is D + (m-n) L. Then at 15 past, the distance from the center to the hands is D + (n-m) L = D - (m-n) L. The distance varies continuously, between D +/- something positive, therefore the result stands.
What did I miss here ?