Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls is odd? (Note that if no die is rolled, their sum is $ 0$.) $ \textbf{(A)}\ \frac{3}{8} \qquad \textbf{(B)}\ \frac{1}{2} \qquad \textbf{(C)}\ \frac{43}{72} \qquad \textbf{(D)}\ \frac{5}{8} \qquad \textbf{(E)}\ \frac{2}{3}$

Juan rolls a fair regular octahedral die marked with the numbers $ 1$ through $ 8$. Then Amal rolls a fair six-sided die. What is the probability that the product of the two rolls is a multiple of $ 3$? $ \textbf{(A)}\ \frac{1}{12} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{1}{2} \qquad \textbf{(D)}\ \frac{7}{12} \qquad \textbf{(E)}\ \frac{2}{3}$

Players $A$ and $B$ play the following game. Each of them throws a dice, and if the outcomes are $x$ and $y$ respectively, a list of all two digit numbers $10a + b$ with $a,b\in \{1,..,6\}$ and $10a + b \le 10x + y$ is created. Then the players alternately reduce the list by replacing a pair of numbers in the list by their absolute difference, until only one number remains. If the remaining number is of the same parity as the outcome of $A$’s throw, then $A$ is proclaimed the winner. What is the probability that $A$ wins the game?

A tree $T$ is given. Starting with the complete graph on $n$ vertices, subgraphs isomorphic to $T$ are erased at random until no such subgraph remains. For what trees does there exist a positive constant $c$ such that the expected number of edges remaining is at least $cn^2$ for all positive integers $n$? [i]David Yang.[/i]

A permutation $ \pi$ is selected randomly through all $ n$-permutations. a) if \[ C_a(\pi)\equal{}\mbox{the number of cycles of length }a\mbox{ in }\pi\] then prove that $ E(C_a(\pi))\equal{}\frac1a$ b) Prove that if $ \{a_1,a_2,\dots,a_k\}\subset\{1,2,\dots,n\}$ the probability that $ \pi$ does not have any cycle with lengths $ a_1,\dots,a_k$ is at most $ \frac1{\sum_{i\equal{}1}^ka_i}$

Six cities $A, B, C, D, E$, and $F$ are located on the vertices of a regular hexagon in that order. $G$ is the center of the hexagon. The sides of the hexagon are the roads connecting these cities. Further more, there are roads connecting cities $B, C, E, F$ and $G$, respectively. Because of raining, one or more roads maybe destroyed. The probability of the road keeping undestroyed between two consecutive cities is $p$. Determine the probability of the road between cities $A$ and $D$ is undestroyed.

A number $ x$ is uniformly chosen on the interval $ [0,1]$, and $ y$ is uniformly randomly chosen on $ [\minus{}1,1]$. Find the probability that $ x>y$.

Suppose that the function $h:\mathbb{R}^2\to\mathbb{R}$ has continuous partial derivatives and satisfies the equation \[h(x,y)=a\frac{\partial h}{\partial x}(x,y)+b\frac{\partial h}{\partial y}(x,y)\] for some constants $a,b.$ Prove that if there is a constant $M$ such that $|h(x,y)|\le M$ for all $(x,y)$ in $\mathbb{R}^2,$ then $h$ is identically zero.

Ali, Bonnie, Carlo, and Dianna are going to drive together to a nearby theme park. The car they are using has $4$ seats: $1$ Driver seat, $1$ front passenger seat, and $2$ back passenger seat. Bonnie and Carlo are the only ones who know how to drive the car. How many possible seating arrangements are there? $\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 24$

A set of lines in the plane is in [i]general position[/i] if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite area; we call these its [i]finite regions[/i]. Prove that for all sufficiently large $n$, in any set of $n$ lines in general position it is possible to colour at least $\sqrt{n}$ lines blue in such a way that none of its finite regions has a completely blue boundary. [i]Note[/i]: Results with $\sqrt{n}$ replaced by $c\sqrt{n}$ will be awarded points depending on the value of the constant $c$.

Evan writes a random positive integer on a board: the integer $k$ has probability $2^{-k}$ of being written. He keeps writing integers in this way repeatedly until he writes an integer that he had written before. He then takes all the integers he has written besides his last, sorts them in the order he first drew them, and also sorts them in increasing order, forming two sequences. For example, if he wrote $5,8,2,3,6,10,2$ in that order then his two sequences would be $5,8,2,3,6,10$ and $2,3,5,6,8,10.$ Find the probability that for all $k \in \{ 1,4,34 \},$ that $k$ was written, and $k$ appears in the same position in both sequences.

Let $X_1$, $X_2$, ..., $X_{2012}$ be chosen independently and uniformly at random from the interval $(0,1]$. In other words, for each $X_n$, the probability that it is in the interval $(a,b]$ is $b-a$. Compute the probability that $\lceil\log_2 X_1\rceil+\lceil\log_4 X_2\rceil+\cdots+\lceil\log_{1024} X_{2012}\rceil$ is even. (Note: For any real number $a$, $\lceil a \rceil$ is defined as the smallest integer not less than $a$.)

Michael draws $\triangle ABC$ in the sand such that $\angle ACB=90^\circ$ and $\angle CBA=15^\circ$. He then picks a point at random from within the triangle and labels it point $M$. Next, he draws a segment from $A$ to $BC$ that passes through $M$, hitting $BC$ at a point he labels $D$. Just then, a wave washes over his work, so Michael redraws the exact same diagram farther from the water, labeling all the points the same way as before. If hypotenuse $AB$ is $4$ feet in length, let $p$ be the probability that the number of feet in the length of $AD$ is less than $2\sqrt3-2$. Compute $\lfloor1000p\rfloor$.

Consider a computer network consisting of servers and bi-directional communication channels among them. Unfortunately, not all channels operate. Each direction of each channel fails with probability $p$ and operates otherwise. (All of these stochastic events are mutually independent, and $0 \le p \le 1$.) There is a root serve, denoted by $r$. We call the network [i]operational[/i], if all serves can reach $r$ using only operating channels. Note that we do not require $r$ to be able to reach any servers. Show that the probability of the network to be operational does not depend on the choice of $r$. (In other words, for any two distinct root servers $r_1$ and $r_2$, the operational probability is the same.)

We are given a coin of diameter $\frac{1}{2}$ and a checkerboard of $1\times1$ squares of area $2010\times2010$. We toss the coin such that it lands completely on the checkerboard. If the probability that the coin doesn't touch any of the lattice lines is $\frac{a^2}{b^2}$ where $\frac{a}{b}$ is a reduced fraction, find $a+b$

If two dice are tossed, the probability that the product of the numbers showing on the tops of the dice is greater than $10$ is $\text{(A)}\ \dfrac{3}{7} \qquad \text{(B)}\ \dfrac{17}{36} \qquad \text{(C)}\ \dfrac{1}{2} \qquad \text{(D)}\ \dfrac{5}{8} \qquad \text{(E)}\ \dfrac{11}{12}$

Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\tfrac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\tfrac{4}{5}$ as $N$ grows large. What is the sum of the digits of the least value of $N$ such that $P(N) < \tfrac{321}{400}$? $\textbf{(A) } 12 \qquad \textbf{(B) } 14 \qquad \textbf{(C) }16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 20$

Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every $ 90$ seconds, and Robert runs clockwise and completes a lap every $ 80$ seconds. Both start from the start line at the same time. At some random time between $ 10$ minutes and $ 11$ minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture? $ \textbf{(A)}\ \frac{1}{16}\qquad \textbf{(B)}\ \frac18\qquad \textbf{(C)}\ \frac{3}{16} \qquad \textbf{(D)}\ \frac14\qquad \textbf{(E)}\ \frac{5}{16}$

Let $ y_0$ be chosen randomly from $ \{0, 50\}$, let $ y_1$ be chosen randomly from $ \{40, 60, 80\}$, let $ y_2$ be chosen randomly from $ \{10, 40, 70, 80\}$, and let $ y_3$ be chosen randomly from $ \{10, 30, 40, 70, 90\}$. (In each choice, the possible outcomes are equally likely to occur.) Let $ P$ be the unique polynomial of degree less than or equal to $ 3$ such that $ P(0) \equal{} y_0$, $ P(1) \equal{} y_1$, $ P(2) \equal{} y_2$, and $ P(3) \equal{} y_3$. What is the expected value of $ P(4)$?

A regular $2004$-sided polygon is given, with all of its diagonals drawn. After some sides and diagonals are removed, every vertex has at most five segments coming out of it. Prove that one can color the vertices with two colors such that at least $\frac{3}{5}$ of the remaining segments have ends with different colors.

[b]10.[/b] Consider a point performing a random walk on a planar triangular lattice and suppose that it moves away with equal probability from any lattice point along any one of the six lattice lines issuing from it. Prove that if the walk is continued indefinitely, then the point will return to its starting point with probability 1. [b](P. 5)[/b]

Ben is throwing darts at a circular target with diameter 10. Ben never misses the target when he throws a dart, but he is equally likely to hit any point on the target. Ben gets $\lceil 5-x \rceil$ points for having the dart land $x$ units away from the center of the target. What is the expected number of points that Ben can earn from throwing a single dart? (Note that $\lceil y \rceil$ denotes the smallest integer greater than or equal to $y$.)

This is an ARML Super Relay! I'm sure you know how this works! You start from #1 and #15 and meet in the middle. We are going to require you to solve all $15$ problems, though -- so for the entire task, submit the sum of all the answers, rather than just the answer to #8. Also, uhh, we can't actually find the slip for #1. Sorry about that. Have fun anyways! Problem 2. Let $T = TNYWR$. Find the number of way to distribute $6$ indistinguishable pieces of candy to $T$ hungry (and distinguishable) schoolchildren, such that each child gets at most one piece of candy. Problem 3. Let $T = TNYWR$. If $d$ is the largest proper divisor of $T$, compute $\frac12 d$. Problem 4. Let $T = TNYWR$ and flip $4$ fair coins. Suppose the probability that at most $T$ heads appear is $\frac mn$, where $m$ and $n$ are coprime positive integers. Compute $m+n$. Problem 5. Let $T = TNYWR$. Compute the last digit of $T^T$ in base $10$. Problem 6. Let $T = TNYWR$ and flip $6$ fair coins. Suppose the probability that at most $T$ heads appear is $\frac mn$, where $m$ and $n$ are coprime positive integers. Compute $m+n$. Problem 7. Let $T = TNYWR$. Compute the smallest prime $p$ for which $n^T \not\equiv n \pmod{p}$ for some integer $n$. Problem 8. Let $M$ and $N$ be the two answers received, with $M \le N$. Compute the number of integer quadruples $(w,x,y,z)$ with $w+x+y+z = M \sqrt{wxyz}$ and $1 \le w,x,y,z \le N$. Problem 9. Let $T = TNYWR$. Compute the smallest integer $n$ with $n \ge 2$ such that $n$ is coprime to $T+1$, and there exists positive integers $a$, $b$, $c$ with $a^2+b^2+c^2 = n(ab+bc+ca)$. Problem 10. Let $T = TNYWR$ and flip $10$ fair coins. Suppose the probability that at most $T$ heads appear is $\frac mn$, where $m$ and $n$ are coprime positive integers. Compute $m+n$. Problem 11. Let $T = TNYWR$. Compute the last digit of $T^T$ in base $10$. Problem 12. Let $T = TNYWR$ and flip $12$ fair coins. Suppose the probability that at most $T$ heads appear is $\frac mn$, where $m$ and $n$ are coprime positive integers. Compute $m+n$. Problem 13. Let $T = TNYWR$. If $d$ is the largest proper divisor of $T$, compute $\frac12 d$. Problem 14. Let $T = TNYWR$. Compute the number of way to distribute $6$ indistinguishable pieces of candy to $T$ hungry (and distinguishable) schoolchildren, such that each child gets at most one piece of candy. Also, we can't find the slip for #15, either. We think the SFBA coaches stole it to prevent us from winning the Super Relay, but that's not going to stop us, is it? We have another #15 slip that produces an equivalent answer. Here you go! Problem 15. Let $A$, $B$, $C$ be the answers to #8, #9, #10. Compute $\gcd(A,C) \cdot B$.

Flora the frog starts at $0$ on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance $m$ with probability $\frac{1}{2^m}$. What is the probability that Flora will eventually land at $10$? $\textbf{(A) } \frac{5}{512} \qquad \textbf{(B) } \frac{45}{1024} \qquad \textbf{(C) } \frac{127}{1024} \qquad \textbf{(D) } \frac{511}{1024} \qquad \textbf{(E) } \frac{1}{2}$

$32$ teams, ranked $1$ through $32$, enter a basketball tournament that works as follows: the teams are randomly paired and in each pair, the team that loses is out of the competition. The remaining $16$ teams are randomly paired, and so on, until there is a winner. A higher ranked team always wins against a lower-ranked team. If the probability that the team ranked $3$ (the third-best team) is one of the last four teams remaining can be written in simplest form as $\dfrac{m}{n}$, compute $m+n$.