HMMT November 2012

What is the sum of all of the distinct prime factors of $25^{3}-27^{2}$ ?

Let $Q(x)=x^{2}+2 x+3$, and suppose that $P(x)$ is a polynomial such that

$$P(Q(x))=x^{6}+6 x^{5}+18 x^{4}+32 x^{3}+35 x^{2}+22 x+8$$

Compute $P(2)$.

$A B C D$ is a rectangle with $A B=20$ and $B C=3$. A circle with radius 5 , centered at the midpoint of $D C$, meets the rectangle at four points: $W, X, Y$, and $Z$. Find the area of quadrilateral $W X Y Z$.

If you roll four fair 6 -sided dice, what is the probability that at least three of them will show the same value?

How many ways are there to arrange three indistinguishable rooks on a $6 \times 6$ board such that no two rooks are attacking each other? (Two rooks are attacking each other if and only if they are in the same row or the same column.)

$A B C D$ is a parallelogram satisfying $A B=7, B C=2$, and $\angle D A B=120^{\circ}$. Parallelogram $E C F A$ is contained in $A B C D$ and is similar to it. Find the ratio of the area of $E C F A$ to the area of $A B C D$.

Find the number of ordered 2012-tuples of integers $\left(x_{1}, x_{2}, \ldots, x_{2012}\right)$, with each integer between 0 and 2011 inclusive, such that the sum $x_{1}+2 x_{2}+3 x_{3}+\cdots+2012 x_{2012}$ is divisible by 2012 .

Let $n$ be the 200th smallest positive real solution to the equation $x-\frac{\pi}{2}=\tan x$. Find the greatest integer that does not exceed $\frac{n}{2}$.

Consider triangle $A B C$ where $B C=7, C A=8$, and $A B=9 . D$ and $E$ are the midpoints of $B C$ and $C A$, respectively, and $A D$ and $B E$ meet at $G$. The reflection of $G$ across $D$ is $G^{\prime}$, and $G^{\prime} E$ meets $C G$ at $P$. Find the length $P G$.

Let $\alpha$ and $\beta$ be reals. Find the least possible value of

$$(2 \cos \alpha+5 \sin \beta-8)^{2}+(2 \sin \alpha+5 \cos \beta-15)^{2} .$$

If $4^{4^{4}}=\sqrt[128]{2^{2^{2^{n}}}}$, find $n$.

If $x^{x}=2012^{2012^{2013}}$, find $x$.

Find the smallest positive integer $n$ such that $\underbrace{2^{2 \cdot 2}}_{n}>3^{3^{3^{3}}}$. (The notation $\underbrace{2^{2 \cdot 2^{2}}}_{n}$ is used to denote a power tower with $n 2$ 's. For example, $\underbrace{2^{2^{2}}}_{n}$ with $n=4$ would equal $2^{2^{2^{2}}}$.)

Find the sum of all real solutions for $x$ to the equation $\left(x^{2}+2 x+3\right)^{\left(x^{2}+2 x+3\right)^{\left(x^{2}+2 x+3\right)}}=2012$.

Given any positive integer, we can write the integer in base 12 and add together the digits of its base 12 representation. We perform this operation on the number $7^{6^{5^{3^{2^{1}}}}}$ repeatedly until a single base 12 digit remains. Find this digit.

A rectangular piece of paper with vertices $A B C D$ is being cut by a pair of scissors. The pair of scissors starts at vertex $A$, and then cuts along the angle bisector of $D A B$ until it reaches another edge of the paper. One of the two resulting pieces of paper has 4 times the area of the other piece. What is the ratio of the longer side of the original paper to the shorter side?

The game of rock-scissors is played just like rock-paper-scissors, except that neither player is allowed to play paper. You play against a poorly-designed computer program that plays rock with $50 \%$ probability and scissors with $50 \%$ probability. If you play optimally against the computer, find the probability that after 8 games you have won at least 4 .

In the game of rock-paper-scissors-lizard-Spock, rock defeats scissors and lizard, paper defeats rock and Spock, scissors defeats paper and lizard, lizard defeats paper and Spock, and Spock defeats rock and scissors, as shown in the below diagram. As before, if two players choose the same move, then there is a draw. If three people each play a game of rock-paper-scissors-lizard-Spock at the same time by choosing one of the five moves at random, what is the probability that one player beats the other two?

64 people are in a single elimination rock-paper-scissors tournament, which consists of a 6-round knockout bracket. Each person has a different rock-paper-scissors skill level, and in any game, the person with the higher skill level will always win. For how many players $P$ is it possible that $P$ wins the first four rounds that he plays? (A 6-round knockout bracket is a tournament which works as follows: (a) In the first round, all 64 competitors are paired into 32 groups, and the two people in each group play each other. The winners advance to the second round, and the losers are eliminated. (b) In the second round, the remaining 32 players are paired into 16 groups. Again, the winner of each group proceeds to the next round, while the loser is eliminated. (c) Each round proceeds in a similar way, eliminating half of the remaining players. After the sixth round, only one player will not have been eliminated. That player is declared the champion.)

In a game of rock-paper-scissors with $n$ people, the following rules are used to determine a champion: (a) In a round, each person who has not been eliminated randomly chooses one of rock, paper, or scissors to play. (b) If at least one person plays rock, at least one person plays paper, and at least one person plays scissors, then the round is declared a tie and no one is eliminated. If everyone makes the same move, then the round is also declared a tie. (c) If exactly two moves are represented, then everyone who made the losing move is eliminated from playing in all further rounds (for example, in a game with 8 people, if 5 people play rock and 3 people play scissors, then the 3 who played scissors are eliminated). (d) The rounds continue until only one person has not been eliminated. That person is declared the champion and the game ends.

If a game begins with 4 people, what is the expected value of the number of rounds required for a champion to be determined?

Find the number of integers between 1 and 200 inclusive whose distinct prime divisors sum to 16 . (For example, the sum of the distinct prime divisors of 12 is $2+3=5$.)

Find the number of ordered triples of divisors $\left(d_{1}, d_{2}, d_{3}\right)$ of 360 such that $d_{1} d_{2} d_{3}$ is also a divisor of 360 .

Find the largest integer less than 2012 all of whose divisors have at most two 1 's in their binary representations.

Let $\pi$ be a permutation of the numbers from 2 through 2012. Find the largest possible value of $\log _{2} \pi(2) \cdot \log _{3} \pi(3) \cdots \log _{2012} \pi(2012)$.

Let $\pi$ be a randomly chosen permutation of the numbers from 1 through 2012 . Find the probability that $\pi(\pi(2012))=2012$.

Let $\pi$ be a permutation of the numbers from 1 through 2012. What is the maximum possible number of integers $n$ with $1 \leq n \leq 2011$ such that $\pi(n)$ divides $\pi(n+1)$ ?

Let $A_{1} A_{2} \ldots A_{100}$ be the vertices of a regular 100-gon. Let $\pi$ be a randomly chosen permutation of the numbers from 1 through 100. The segments $A_{\pi(1)} A_{\pi(2)}, A_{\pi(2)} A_{\pi(3)}, \ldots, A_{\pi(99)} A_{\pi(100)}, A_{\pi(100)} A_{\pi(1)}$ are drawn. Find the expected number of pairs of line segments that intersect at a point in the interior of the 100-gon.

$A B C$ is a triangle with $A B=15, B C=14$, and $C A=13$. The altitude from $A$ to $B C$ is extended to meet the circumcircle of $A B C$ at $D$. Find $A D$.

Triangle $A B C$ satisfies $\angle B>\angle C$. Let $M$ be the midpoint of $B C$, and let the perpendicular bisector of $B C$ meet the circumcircle of $\triangle A B C$ at a point $D$ such that points $A, D, C$, and $B$ appear on the circle in that order. Given that $\angle A D M=68^{\circ}$ and $\angle D A C=64^{\circ}$, find $\angle B$.

Triangle $A B C$ has $A B=4, B C=5$, and $C A=6$. Points $A^{\prime}, B^{\prime}, C^{\prime}$ are such that $B^{\prime} C^{\prime}$ is tangent to the circumcircle of $\triangle A B C$ at $A, C^{\prime} A^{\prime}$ is tangent to the circumcircle at $B$, and $A^{\prime} B^{\prime}$ is tangent to the circumcircle at $C$. Find the length $B^{\prime} C^{\prime}$.