4th Annual Harvard-MIT November Tournament Theme Round Problems

4th Annual Harvard-MIT November Tournament
Saturday 12 November 2011
Theme Round
Fish
1. [3] Five of James’ friends are sitting around a circular table to play a game of Fish. James chooses a
place between two of his friends to pull up a chair and sit. Then, the six friends divide themselves into
two disjoint teams, with each team consisting of three consecutive players at the table. If the order in
which the three members of a team sit does not matter, how many possible (unordered) pairs of teams
are possible?
2. [3] In a game of Fish, R2 and R3 are each holding a positive number of cards so that they are collectively
holding a total of 24 cards. Each player gives an integer estimate for the number of cards he is holding,
such that each estimate is an integer between 80% of his actual number of cards and 120% of his actual
number of cards, inclusive. Find the smallest possible sum of the two estimates.
3. [5] In preparation for a game of Fish, Carl must deal 48 cards to 6 players. For each card that he deals,
he runs through the entirety of the following process:
1. He gives a card to a random player.
2. A player Z is randomly chosen from the set of players who have at least as many cards as every
other player (i.e. Z has the most cards or is tied for having the most cards).
3. A player D is randomly chosen from the set of players other than Z who have at most as many
cards as every other player (i.e. D has the fewest cards or is tied for having the fewest cards).
4. Z gives one card to D.
He repeats steps 1-4 for each card dealt, including the last card. After all the cards have been dealt,
what is the probability that each player has exactly 8 cards?
4. [6] Toward the end of a game of Fish, the 2 through 7 of spades, inclusive, remain in the hands of three
distinguishable players: DBR, RB, and DB, such that each player has at least one card. If it is known
that DBR either has more than one card or has an even-numbered spade, or both, in how many ways
can the players’ hands be distributed?
5. [8] For any finite sequence of positive integers ¼, let S(¼) be the number of strictly increasing sub-
sequences in ¼ with length 2 or more. For example, in the sequence ¼ = {3, 1, 2, 4}, there are five
increasing sub-sequences: {3, 4}, {1, 2}, {1, 4}, {2, 4}, and {1, 2, 4}, so S(¼) = 5. In an eight-player
game of Fish, Joy is dealt six cards of distinct values, which she puts in a random order ¼ from left to
right in her hand. Determine
X
¼
S(¼),
where the sum is taken over all possible orders ¼ of the card values.
Circles and Tangents
6. [3] Let ABC be an equilateral triangle with AB = 3. Circle ! with diameter 1 is drawn inside the
triangle such that it is tangent to sides AB and AC. Let P be a point on ! and Q be a point on
segment BC. Find the minimum possible length of the segment PQ.
7. [4] Let XY Z be a triangle with \XY Z = 40± and \Y ZX = 60±. A circle ¡, centered at the point
I, lies inside triangle XY Z and is tangent to all three sides of the triangle. Let A be the point of
tangency of ¡ with Y Z, and let ray
−!
XI intersect side Y Z at B. Determine the measure of \AIB.
8. [5] Points D,E, F lie on circle O such that the line tangent to O at D intersects ray
−−!
EF at P. Given
that PD = 4, PF = 2, and \FPD = 60±, determine the area of circle O.
9. [6] Let ABC be a triangle with AB = 13, BC = 14, and CA = 15. Let D be the foot of the altitude
from A to BC. The inscribed circles of triangles ABD and ACD are tangent to AD at P and Q,
respectively, and are tangent to BC at X and Y , respectively. Let PX and QY meet at Z. Determine
the area of triangle XY Z.
10. [7] Let ­ be a circle of radius 8 centered at point O, and let M be a point on ­. Let S be the set of
points P such that P is contained within ­, or such that there exists some rectangle ABCD containing
P whose center is on ­ with AB = 4, BC = 5, and BC k OM. Find the area of S.