Harvard Wins Math Contest

These are two of six questions from the first three-hour section of the fifty-eighth annual William Lowell Putnam Mathematical Competition. A second section of similar length followed.

1 A rectangle,HOMF, has sides HO =11 and OM =5. A triangle ABC has H as the intersection of the attitudes, O the center of the circumscribed circle, M the midpoint of BC, and F the foot of the altitude from A. What is the length of BC?

2 Players 1,2,3, ..., n are seated around a table and each has a single penny. Player 1 penny to player 2, who then passes two pennies to player 3. Player 3 then passes one penny to player 4, who passes two pennies to player 5, and so on, players alternately passing one penny or two to the next player who still has some pennies. A player who runs out of pennies drops out of the game and leaves the table. Find an infinite set of numbers n for which some player ends up with all n pennies.

Source: William Lowell Putnam Mathematical Competition 1997, Examination A