MathDB

4

Part of 2016 Harvard-MIT Mathematics Tournament

Problems(7)

2016 Algebra #4

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12/24/2016
Determine the remainder when ∑i=02015⌊2i25⌋\sum_{i=0}^{2015} \left\lfloor \frac{2^i}{25} \right\rfloor is divided by 100, where ⌊x⌋\lfloor x \rfloor denotes the largest integer not greater than xx.
2016 Combo #4

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12/30/2016
Let RR be the rectangle in the Cartesian plane with vertices at (0,0),(2,0),(2,1),(0,0), (2,0), (2,1), and (0,1)(0,1). RR can be divided into two unit squares, as shown; the resulting figure has seven edges.
[asy] size(3cm); draw((0,0)--(2,0)--(2,1)--(0,1)--cycle); draw((1,0)--(1,1)); [/asy] How many subsets of these seven edges form a connected figure?
2016 Geo #4

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12/30/2016
Let ABCABC be a triangle with AB=3AB = 3, AC=8AC = 8, BC=7BC = 7 and let MM and NN be the midpoints of AB‾\overline{AB} and AC‾\overline{AC}, respectively. Point TT is selected on side BCBC so that AT=TCAT = TC. The circumcircles of triangles BATBAT, MANMAN intersect at DD. Compute DCDC.
2016 Guts #4

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12/24/2016
Consider a three-person game involving the following three types of fair six-sided dice.
    All three players simultaneously choose a die (more than one person can choose the same type of die, and the players don't know one another's choices) and roll it. Then the score of a player PP is the number of players whose roll is less than PP's roll (and hence is either 00, 11, or 22). Assuming all three players play optimally, what is the expected score of a particular player?
    2016 Team #4

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    12/30/2016
    Let n>1n > 1 be an odd integer. On an n×nn \times n chessboard the center square and four corners are deleted. We wish to group the remaining n2−5n^2-5 squares into 12(n2−5)\frac12(n^2-5) pairs, such that the two squares in each pair intersect at exactly one point (i.e.\ they are diagonally adjacent, sharing a single corner).
    For which odd integers n>1n > 1 is this possible?
    2016 General #4: Playing pool

    Source:

    11/15/2016
    A rectangular pool table has vertices at (0,0)(12,0)(0,10),(0, 0) (12, 0) (0, 10), and (12,10)(12, 10). There are pockets only in the four corners. A ball is hit from (0,0)(0, 0) along the line y=xy = x and bounces off several walls before eventually entering a pocket. Find the number of walls that the ball bounces off of before entering a pocket.
    HMMT
    2016 Theme #4: Square vertices as points

    Source:

    11/22/2016
    A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these 44 numbers?
    HMMT