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Problem of the Year

Year 2010
Consider the card game 'the war' for two players:  a well shuffled deck of 52 cards is divided into two.
Each player turns over the card on the top and the person with a larger card wins: Ace is the largest and 2 is
the smallest. The winner then put the cards at the bottom of his stack and starts the next round.
If two cards are the same kind, both players continue to turn over the top cards and compare their 4th cards.
The game ends when one of the players does not have any cards in his hand.
Q1: In what kind of situations (for which order of cards), this game will never end?
Q2: Consider all the situations where the game does end. Find the average number of rounds two players need to
play before the game ends.
Q3: Find the largest possible number of rounds (finite) of the game.
Q4. If  we use a regular PC to play all the games which end in finite rounds, approximately, how much time do we need to

Year 2009
The numbers 209 and 20009 are divisible by 11: 11 X 19=209, 11 x 1819=20009. Indeed,
the number 2 x 10^n + 9 is divisible by 11 if n is even. Determine whether 2 x 10^n + 9
is a prime number when n is an odd positive integer.