There are 100 old(non-digital) watches in an antique shop, all running but not necessarily on time. Prove that at some moment of time the sum of the distances from the center of the shop to the center of the watches will be less than the sum of the distances from the center of the shop to the ends of the hour hands of the watches.
Will this remain true if if some of them are running fast or slow?
Hint: note that for a given watch, the hour hand tip is on average further from the center of the shop than the center.
For the second part, it still works if you wait long enough for the watches to come back together if the periods are commeasurable-the LCM of the periods, or for the hands to equidistribute if they are not commeasurable.