A curious "Proof Without Words" solution exists for the puzzle at the back cover of Mathematical Puzzles: A Connoisseur's Collection (163 pages, 2003) by Peter Winkler. You might want to pause and solve the problem before scrolling down to see the solution.
2) Putnam 2004 Problem (PDF): "Basketball star Shanille O-Keal's team statistician keeps track of the number S(N) of successful free throws she has made in her first N attempts of the season. Early in the season, S(N) was less than 80% of N, but by the end of the season, S(N) was more than 80%. Was there necessarily a moment in between when S(N) was exactly 80% of N?"