1. n≥1. 2n pirates – Captain Jack and his 2n-1 crewmen – sit in a circle to split a treasure of k gold coins. Jack must decide how many coins to take for himself and how many to give each crewman (not necessarily the same number to each). The 2n-1 crewmen will then vote on Jack's decision. Each is greedy and will vote Yea only if he gets more coins than each of his two neighbors. If a majority vote Yea, Jack's decision is accepted. Otherwise Jack is thrown overboard and gets nothing.  What is the most coins Captain Jack can take for himself and survive?

2. Rose and Bella take turns painting cells on an infinite piece of graph paper red and blue. On Rose's turn, she picks any 2015 blank cells and paints them red. Bella, on her turn, picks any blank cell and paints it blue. Bella wins if the paper has four blue cells arranged as corners of a square of any size with sides parallel to the grid lines. Rose goes first. Show that she cannot prevent Bella from winning.

3. A 2015x2015x2015 cube is divided into smaller cubical rooms by 1x1 square dividers. Show that the number of dividers is divisible by 6. (Generalize to n-dimensional cube.)

4. Each robot in the Martian Army is equipped with a battery that lasts some number of hours. A robot works until its battery is depleted, then recharges its battery until it is full, then goes back to work, and so on. A battery that lasts N hours takes exactly N hours to recharge. We order the robots by descending battery life: a[1], a[2], ..., a[k].  Under what conditions on the a[i] will there necessarily be a moment in time when all the robots are recharging (so you can invade the planet)? For example, it is sufficient that a[i]≥3a[i+1]. Is that condition necessary? Show that the necessary and sufficient condition is a[1]≥2a[2]+4a[3]+8a[4]+...+2^na[n].

5. Read about the so-called "Pumping Lemma" for context-free grammars = nondeterministic finite-state automata. Then consider a casino machine that accepts tokens of 32 different colors, one at a time. For each color, the player can choose between two fixed rewards. Each reward is up to $10 cash, plus maybe another token. For example, a blue token always gives the player a choice of getting either $5 plus a red token or $3 plus a yellow token; a black token can always be exchanged either for $10 (but no token) or for a brown token (but no cash). A player may keep playing as long as he has a token. Rob and Bob each have one white token. Rob watches Bob play and win $500. Use the Pumping Lemma to show that Rob can win at least $1000000.

6. Use p-adic valuations to show that if a set of 2015 rational numbers has a sum that is an integer and the product of any two of them is an integer, then they are all integers. (Recall that the p-adic valuation of the product of two numbers is the product of their valuations, and the valuation of the product of two numbers is the maximum of their valuations. These valuations can be extended to all real numbers, a fact we stated without proof. But the statement is false for a set of irrationals, like 2+sqrt(2) and 2-sqrt(2), whose product and sum are both integers. Where does the proof fail?)

7. Fill in the ?? with the largest number possible and prove: An N×N×N cube is filled with integers such that numbers in cells that share a face differ by at most 1. Then there is some number that appears in the table at least ?? times. Generalize to a d-dimensional cube. Is the answer is (N+d-2)choose(d-1)?