Suppose we choose two numbers $x,y\in[0,1]$ uniformly at random. If the probability that the circle with center $(x,y)$ and radius $|x-y|$ lies entirely within the unit square $[0,1]\times [0,1]$ is written as $\tfrac{p}{q}$ with $p$ and $q$ relatively prime nonnegative integers, then what is $p^2+q^2$?

Each face of a cube is painted either red or blue, each with probability $ 1/2$. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color? $ \textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac{5}{16} \qquad \textbf{(C)}\ \frac38 \qquad \textbf{(D)}\ \frac{7}{16} \qquad \textbf{(E)}\ \frac12$

Let $ \{A_n\}_{n \in \mathbb{N}}$ be a sequence of measurable subsets of the real line which covers almost every point infinitely often. Prove, that there exists a set $ B \subset \mathbb{N}$ of zero density, such that $ \{A_n\}_{n \in B}$ also covers almost every point infinitely often. (The set $ B \subset \mathbb{N}$ is of zero density if $ \lim_{n \to \infty} \frac {\#\{B \cap \{0, \dots, n \minus{} 1\}\}}{n} \equal{} 0$.)

For a permutation $\pi$ of $\{1,2,3,\ldots,n\}$, let $\text{Inv}(\pi)$ be the number of pairs $(i,j)$ with $1 \leq i < j \leq n$ and $\pi(i) > \pi(j)$. [list=1] [*] Given $n$, what is $\sum \text{Inv}(\pi)$ where the sum ranges over all permutations $\pi$ of $\{1,2,3,\ldots,n\}$? [*] Given $n$, what is $\sum \left(\text{Inv}(\pi)\right)^2$ where the sum ranges over all permutations $\pi$ of $\{1,2,3,\ldots,n\}$?[/list] [i]Brian Hamrick.[/i]

Eight teams are competing in a tournament all against all (every pair of team play exactly one time among them). There are not ties and both results of every game are equally probable. What is the probability that in the tournament every team had lose at least one game and won at least one game?

A fair die is rolled four times. The probability that each of the final three rolls is at least as large as the roll preceding it may be expressed in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Suppose we wish to pick a random integer between $1$ and $N$ inclusive by flipping a fair coin. One way we can do this is through generating a random binary decimal between $0$ and $1$, then multiplying the result by $N$ and taking the ceiling. However, this would take an infinite amount of time. We therefore stopping the flipping process after we have enough flips to determine the ceiling of the number. For instance, if $N=3$, we could conclude that the number is $2$ after flipping $.011_2$, but $.010_2$ is inconclusive. Suppose $N=2014$. The expected number of flips for such a process is $\frac{m}{n}$ where $m$, $n$ are relatively prime positive integers, find $100m+n$. [i]Proposed by Lewis Chen[/i]

A box contains $2$ red marbles, $2$ green marbles, and $2$ yellow marbles. Carol takes $2$ marbles from the box at random; then Claudia takes $2$ of the remaining marbles at random; and then Cheryl takes the last two marbles. What is the probability that Cheryl gets $2$ marbles of the same color? $\textbf{(A) }\dfrac1{10}\qquad\textbf{(B) }\dfrac16\qquad\textbf{(C) }\dfrac15\qquad\textbf{(D) }\dfrac13\qquad\textbf{(E) }\dfrac12$

Let $S=\{-100,-99,-98,\ldots,99,100\}$. Choose a $50$-element subset $T$ of $S$ at random. Find the expected number of elements of the set $\{|x|:x\in T\}$.

There are $256$ players in a tennis tournament who are ranked from $1$ to $256$, with $1$ corresponding to the highest rank and $256$ corresponding to the lowest rank. When two players play a match in the tournament, the player whose rank is higher wins the match with probability $\frac{3}{5}$. In each round of the tournament, the player with the highest rank plays against the player with the second highest rank, the player with the third highest rank plays against the player with the fourth highest rank, and so on. At the end of the round, the players who win proceed to the next round and the players who lose exit the tournament. After eight rounds, there is one player remaining and they are declared the winner. Determine the expected value of the rank of the winner.

There is a board numbered $-10$ to $10$. Each square is coloured either red or white, and the sum of the numbers on the red squares is $n$. Maureen starts with a token on the square labeled $0$. She then tosses a fair coin ten times. Every time she flips heads, she moves the token one square to the right. Every time she flips tails, she moves the token one square to the left. At the end of the ten flips, the probability that the token finishes on a red square is a rational number of the form $\frac a b$. Given that $a + b = 2001$, determine the largest possible value for $n$.

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.)

Let $P(x)=x^2-3x-9$. A real number $x$ is chosen at random from the interval $5\leq x \leq 15$. The probability that $\lfloor \sqrt{P(x)} \rfloor = \sqrt{P(\lfloor x \rfloor )}$ is equal to $\dfrac{\sqrt{a}+\sqrt{b}+\sqrt{c}-d}{e}$, where $a,b,c,d$ and $e$ are positive integers and none of $a,b,$ or $c$ is divisible by the square of a prime. Find $a+b+c+d+e$.

Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a 2-by-2 red square is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Reimu and Sanae play a game using $4$ fair coins. Initially both sides of each coin are white. Starting with Reimu, they take turns to color one of the white sides either red or green. After all sides are colored, the four coins are tossed. If there are more red sides showing up, then Reimu wins, and if there are more green sides showing up, then Sanae wins. However, if there is an equal number of red sides and green sides, then [i]neither[/i] of them wins. Given that both of them play optimally to maximize the probability of winning, what is the probability that Reimu wins?

Janet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal 3? $\textbf{(A) }\frac{2}{9}\qquad\textbf{(B) }\frac{49}{216}\qquad\textbf{(C) }\frac{25}{108}\qquad\textbf{(D) }\frac{17}{72}\qquad\textbf{(E) }\frac{13}{54}$

In the game of Yahtzee , players have to achieve various combinations of values with $5$ dice. In a round, a player can roll the dice three times. At the second and third rolls, he can choose which dice to re-roll and which to keep. What is the probability that a player achieves at least four $6$’s in a round, given that he plays with the optimal strategy to maximise this probability? Writing the answer as $p/q$ where $p$ and $q$ are coprime, you should submit the sum of all prime factors of $p$, counted with multiplicity. So for example if you obtained $\frac{p}{q} = \frac{3^4 \cdot 11}{ 2^5 \cdot 5}$ then the submitted answer should be $4 \cdot 3 + 11 = 23$.

The numbers 1, 2, 3, 4, 5, 6, 7, and 8 are randomly written on the faces of a regular octahedron so that each face contains a different number. The probability that no two consecutive numbers, where 8 and 1 are considered to be consecutive, are written on faces that share an edge is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Does there exist two unfair dices such that probability of their sum being $j$ be a number in $\left(\frac2{33},\frac4{33}\right)$ for each $2\leq j\leq 12$?

A frog starts at $0$ on a number line and plays a game. On each turn the frog chooses at random to jump $1$ or $2$ integers to the right or left. It stops moving if it lands on a nonpositive number or a number on which it has already landed. If the expected number of times it will jump is $\tfrac{p}{q}$ for relatively prime positive integers $p$ and $q$, find $p+q$. [i]Proposed by Michael Kural[/i]

An angle $x$ is chosen at random from the interval $0^\circ < x < 90^\circ$. Let $p$ be the probability that the numbers $\sin^2 x$, $\cos^2 x$, and $\sin x \cos x$ are not the lengths of the sides of a triangle. Given that $p = d/n$, where $d$ is the number of degrees in $\arctan m$ and $m$ and $n$ are positive integers with $m + n < 1000$, find $m + n$.

Three points are randomly placed on a circle. What is the probability that they lie on the same semicircle

Positive integers $ a,b,$ and $ c$ are randomly and independently selected with replacement from the set $ \{ 1,2,3,\dots,2010 \}.$ What is the probability that $ abc \plus{} ab \plus{} a$ is divisible by $ 3$? $ \textbf{(A)}\ \dfrac{1}{3} \qquad\textbf{(B)}\ \dfrac{29}{81} \qquad\textbf{(C)}\ \dfrac{31}{81} \qquad\textbf{(D)}\ \dfrac{11}{27} \qquad\textbf{(E)}\ \dfrac{13}{27}$

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 $\gcd(A,C) \cdot B$.

The numbers $1,2,\dots,9$ are randomly placed into the $9$ squares of a $3 \times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd? $\textbf{(A) }1/21\qquad\textbf{(B) }1/14\qquad\textbf{(C) }5/63\qquad\textbf{(D) }2/21\qquad\textbf{(E) } 1/7$