Echo the gecko starts on the point $(0, 0)$ in the 2D coordinate plane. Every minute, starting at the end of the first minute, he'll teleport $1$ unit up, left, right, or down with equal probability. Echo dies the moment he lands on a point that is more than $1$ unit away from the origin. The average number of minutes he'll live can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [i]Team #2[/i]

A box contains 2 pennies, 4 nickels, and 6 dimes. Six coins are drawn without replacement, with each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least 50 cents? $\text{(A)} \ \frac{37}{924} \qquad \text{(B)} \ \frac{91}{924} \qquad \text{(C)} \ \frac{127}{924} \qquad \text{(D)} \ \frac{132}{924} \qquad \text{(E)} \ \text{none of these}$

Irja and Valtteri are tossing coins. They take turns, Irja starting. Each of them has a pebble which reside on opposite vertices of a square at the start. If a player gets heads, she or he moves her or his pebble on opposite vertex. Otherwise the player in turn moves her or his pebble to an adjacent vertex so that Irja proceeds in positive and Valtteri in negative direction. The winner is the one who can move his pebble to the vertex where opponent's pebble lies. What is the probability that Irja wins the game?

In a tennis ball factory there are $4$ machines $m_1 , m_2 , m_3 , m_4$, which produce, respectively, $10\%$, $20\%$, $30\%$ and $40\%$ of the balls that come out of the factory. The machine $m_1$ introduces defects in $1\%$ of the balls it manufactures, the machine $m_2$ in $2\%$, $m_3$ in $4\%$ and $m_4$ in $15\%$. Of the balls manufactured In one day, one is chosen at random and it turns out to be defective. What is the probability that Has this ball been made by the machine $ m_3$ ?

Aumann, Bill, and Charlie each roll a fair $6$-sided die with sides labeled $1$ through $6$ and look at their individual rolls. Each flips a fair coin and, depending on the outcome, looks at the roll of either the player to his right or the player to his left, without anyone else knowing which die he observed. Then, at the same time, each of the three players states the expected value of the sum of the rolls based on the information he has. After hearing what everyone said, the three players again state the expected value of the sum of the rolls based on the information they have. Then, for the third time, after hearing what everyone said, the three players again state the expected value of the sum of the rolls based on the information they have. Prove that Aumann, Bill, and Charlie say the same number the third time.

In a test, $3n$ students participate, who are located in three rows of $n$ students in each. The students leave the test room one by one. If $N_1(t), N_2(t), N_3(t)$ denote the numbers of students in the first, second, and third row respectively at time $t$, find the probability that for each t during the test, \[|N_i(t) - N_j(t)| < 2, i \neq j, i, j = 1, 2, \dots .\]

On an infinite chessboard (whose squares are labeled by $ (x, y)$, where $ x$ and $ y$ range over all integers), a king is placed at $ (0, 0)$. On each turn, it has probability of $ 0.1$ of moving to each of the four edge-neighboring squares, and a probability of $ 0.05$ of moving to each of the four diagonally-neighboring squares, and a probability of $ 0.4$ of not moving. After $ 2008$ turns, determine the probability that the king is on a square with both coordinates even. An exact answer is required.

Steve needed to address a letter to $2743$ Becker Road. He remembered the digits of the address, but he forgot the correct order of the digits, so he wrote them down in random order. The probability that Steve got exactly two of the four digits in their correct positions is $\tfrac m n$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let $d(n)$ be the number of positive divisors of a positive integer $n$ (including $1$ and $n$). Find all values of $n$ such that $n + d(n) = d(n)^2$.

An urn contains $n$ balls labeled $1, 2, ... , n$. We draw the balls out one by one (without replacing them) until we obtain a ball whose number is divisible by $k$. Find all $k$ such that the expected number of balls removed is $k$.

In a new school $40$ percent of the students are freshmen, $30$ percent are sophomores, $20$ percent are juniors, and $10$ percent are seniors. All freshmen are required to take Latin, and $80$ percent of the sophomores, $50$ percent of the juniors, and $20$ percent of the seniors elect to take Latin. The probability that a randomly chosen Latin student is a sophomore is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

A point $ P$ is selected at random from the interior of the pentagon with vertices $ A \equal{} (0,2)$, $B \equal{} (4,0)$, $C \equal{} (2 \pi \plus{} 1, 0)$, $D \equal{} (2 \pi \plus{} 1,4)$, and $ E \equal{} (0,4)$. What is the probability that $ \angle APB$ is obtuse? [asy] size(150); pair A, B, C, D, E; A = (0,1.5); B = (3,0); C = (2 *pi + 1, 0); D = (2 * pi + 1,4); E = (0,4); draw(A--B--C--D--E--cycle); label("$A$", A, dir(180)); label("$B$", B, dir(270)); label("$C$", C, dir(0)); label("$D$", D, dir(0)); label("$E$", E, dir(180)); [/asy] $ \displaystyle \textbf{(A)} \ \frac {1}{5} \qquad \textbf{(B)} \ \frac {1}{4} \qquad \textbf{(C)} \ \frac {5}{16} \qquad \textbf{(D)} \ \frac {3}{8} \qquad \textbf{(E)} \ \frac {1}{2}$

Each of 2010 boxes in a line contains a single red marble, and for $ 1 \le k \le 2010$, the box in the $ kth$ position also contains $ k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $ P(n)$ be the probability that Isabella stops after drawing exactly $ n$ marbles. What is the smallest value of $ n$ for which $ P(n) < \frac {1}{2010}$? $ \textbf{(A)}\ 45 \qquad \textbf{(B)}\ 63 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 201 \qquad \textbf{(E)}\ 1005$

A token starts at the point $(0,0)$ of an $xy$-coordinate grid and them makes a sequence of six moves. Each move is $1$ unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other moves. The probability the token ends at a point on the graph of $|y|=|x|$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

$20$ players are playing in a Super Mario Smash Bros. Melee tournament. They are ranked $1-20$, and player $n$ will always beat player $m$ if $n<m$. Out of all possible tournaments where each player plays $18$ distinct other players exactly once, one is chosen uniformly at random. Find the expected number of pairs of players that win the same number of games.

Consider a matrix of size $n\times n$ whose entries are real numbers of absolute value not exceeding $1$. The sum of all entries of the matrix is $0$. Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding $C$ in absolute value. [i]Proposed by Marcin Kuczma, Poland[/i]

Four different positive integers less than 10 are chosen randomly. What is the probability that their sum is odd?

There are $10$ seats in each of $10$ rows of a theatre and all the seats are numbered. What is the probablity that two friends buying tickets independently will occupy adjacent seats? $ \textbf{a)}\ \dfrac{1}{55} \qquad\textbf{b)}\ \dfrac{1}{50} \qquad\textbf{c)}\ \dfrac{2}{55} \qquad\textbf{d)}\ \dfrac{1}{25} \qquad\textbf{e)}\ \text{None of above} $

Real numbers $x,y,$ and $z$ are chosen from the interval $[-1,1]$ independently and uniformly at random. What is the probability that $$\vert{x}\vert+\vert{y}\vert+\vert{z}\vert+\vert{x+y+z}\vert=\vert{x+y}\vert+\vert{y+z}\vert+\vert{z+x}\vert?$$

Consider the L-shaped region formed by three unit squares joined at their sides, as shown below. Two points $A$ and $B$ are chosen independently and uniformly at random from inside this region. The probability that the midpoint of $\overline{AB}$ also lies inside this L-shaped region can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [asy] size(2.5cm); draw((0,0)--(0,2)--(1,2)--(1,1)--(2,1)--(2,0)--cycle); draw((0,1)--(1,1)--(1,0), dotted); [/asy]

Let $ S$ be a set that contains $ n$ elements. Let $ A_{1},A_{2},\cdots,A_{k}$ be $ k$ distinct subsets of $ S$, where $ k\geq 2, |A_{i}| \equal{} a_{i}\geq 1 ( 1\leq i\leq k)$. Prove that the number of subsets of $ S$ that don't contain any $ A_{i} (1\leq i\leq k)$ is greater than or equal to $ 2^n\prod_{i \equal{} 1}^k(1 \minus{} \frac {1}{2^{a_{i}}}).$

Dave arrives at an airport which has twelve gates arranged in a straight line with exactly $ 100$ feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at random. Let the probability that Dave walks $ 400$ feet or less to the new gate be a fraction $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.

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]

Let $S$ be a randomly chosen $6$-element subset of the set $\{0,1,2,\ldots,n\}.$ Consider the polynomial $P(x)=\sum_{i\in S}x^i.$ Let $X_n$ be the probability that $P(x)$ is divisible by some nonconstant polynomial $Q(x)$ of degree at most $3$ with integer coefficients satisfying $Q(0) \neq 0.$ Find the limit of $X_n$ as $n$ goes to infinity.

Lil Wayne, the rain god, determines the weather. If Lil Wayne makes it rain on any given day, the probability that he makes it rain the next day is $75\%$. If Lil Wayne doesn't make it rain on one day, the probability that he makes it rain the next day is $25\%$. He decides not to make it rain today. Find the smallest positive integer $n$ such that the probability that Lil Wayne [i]makes it rain[/i] $n$ days from today is greater than $49.9\%$.