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**1.** Feng and Simona are playing a game in which they take
turns breaking up a rectangular bar of chocolate, six squares wide by
eight squares long. Each break must be in a straight line along the
divisions between the squares, and must go all the way across. Once
the bar breaks into several pieces, the players continue taking turns
choosing one piece and breaking it, until only the individual squares
remain. The player who makes the last break wins the game (and the
chocolate). If Feng goes first, who will win? Is there a strategy
that this player has to follow? Be sure you explain why she can be
assured of her victory, no matter what moves her opponent makes.

**2.** Find the smallest positive integer *n* such that
*n*/2 is the square of an integer, *n*/3 is the cube of an
integer, *n*/5 is the fifth power of an integer, and *n*/11
is the 11th power of an integer.

**3.** Mark is drawing up the Mathcamp schedule, doing his best
to make it possible for everyone to attend the classes of their
choice. 70% of the campers want to take Alice's class on set theory,
75% of the campers want to go to Brenda's class on projective
geometry, 80% of the campers say they're interested in Chris's class
on group theory, and 85% of the campers want to take Dave's number
theory class. What's the smallest possible percentage of campers that
want to go to all four classes?

**4.** On a regular dodecagon (12-sided polygon), number the
vertices 1 through 12 in order, as on the face of a clock. Connect 11
to 3, 8 to 12, and 10 to 1. Do these three lines meet in a single
point? Be sure to prove your answer.

**5.** Three bugs are crawling on the coordinate plane. They move
one at a time, and each bug will only crawl in a direction parallel to
the line joining the other two.

(a) If the bugs start out at (0,0), (3,0), and (0,3), is it possible that
after some time the first bug will end up back where it started, while the
other two bugs switch places?

(b) Can the three bugs end up at (1,2), (2,5), and (-2,3)?

**6.** Nineteen Mathcamp students have signed up for a ping-pong tournament in
which each participant is to play exactly 10 games, and no one should play
the same opponent twice.

(a) Show that it is possible to arrange such a tournament.

(b) Show that any two players who play one another have at least one
common opponent.

(c) Is it possible to arrange the tournament so that any two players who
play one another have exactly one common opponent?

**7.** The Rainbow Game is played by a team of seven. Each
player gets a hat, which can be any one of the seven colors red,
orange, yellow, green, blue, indigo, and violet. The colors of the
hats are independent of each other and repetitions are allowed: for
instance, it may happen that all the hats are green. Each player can
see only the colors of the six hats worn by the rest of the team; no
player can see the color of his or her own hat. The players are to
guess the colors of their own hats, and if at least one player guesses
correctly then the team as a whole wins.

The players may not communicate in any fashion during the game, and they must all announce their guesses simultaneously. They are, however, allowed to plan out a strategy in advance. Is there a strategy for choosing their guesses that will allow the team to win 100% of the time? (As a warm-up, try it with two players and two colors).

**8.** The sequence a_{0} = 0, a_{1} = 1,
a_{2} = 4, a_{3} = 15, ... is defined via the
recursion a_{n} = 4 a_{n-1} - a_{n-2} for n >=
2. Show that the triple of numbers (a_{n-1}, 2 a_{n},
a_{n+1}) has the property that the product of any two of the
numbers is one less than a perfect square.

**9.** The number 1089 has the property that to multiply it by 9 you just
reverse the digits, i.e. 9 x 1089 = 9801. How many n-digit numbers are
there with this property?

**10.** Parity High School has *n* students. The school rules allow
the students to form clubs, but dictate that each club must have an odd
number of members, and that any two clubs must have an even number of
members in common. What is the largest possible number of clubs? Be sure
to prove your answer.

*For those who happen to know advanced linear algebra, there is a very
short and elegant solution to this problem. We suspect that there is an
elementary solution as well, but we have not been able to find it. We are
hoping that one of you can--consider this our final challenge to you!*