p1. Let A(S) denote the average value of a set S. Let T be the set of all subsets of the set {1,2,3,4,...,2012}, and let R be {A(K)∣K∈T}. Compute A(R).
p2. Consider the minute and hour hands of the Campanile, our clock tower. During one single day (12:00 AM - 12:00 AM), how many times will the minute and hour hands form a right-angle at the center of the clock face?
p3. In a regular deck of 52 face-down cards, Billy flips 18 face-up and shuffles them back into the deck. Before giving the deck to Penny, Billy tells her how many cards he has flipped over, and blindfolds her so she can’t see the deck and determine where the face-up cards are. Once Penny is given the deck, it is her job to split the deck into two piles so that both piles contain the same number of face-up cards. Assuming that she knows how to do this, how many cards should be in each pile when he is done?
p4. The roots of the equation x3+ax2+bx+c=0 are three consecutive integers. Find the maximum value of b+1a2.
p5. Oski has a bag initially filled with one blue ball and one gold ball. He plays the following game: first, he removes a ball from the bag. If the ball is blue, he will put another blue ball in the bag with probability 4371 and a gold ball in the bag the rest of the time. If the ball is gold, he will put another gold ball in the bag with probability 4371 and a blue ball in the bag the rest of the time. In both cases, he will put the ball he drew back into the bag. Calculate the expected number of blue balls after 525600 iterations of this game.
p6. Circles A and B intersect at points C and D. Line AC and circle B meet at E, line BD and circle A meet at F, and lines EF and AB meet at G. If AB=10, EF=4, FG=8, find BG.
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