A particle moves from $(0, 0)$ to $(n, n)$ directed by a fair coin. For each head it moves one step east and for each tail it moves one step north. At $(n, y), y < n$, it stays there if a head comes up and at $(x, n), x < n$, it stays there if a tail comes up. Let$ k$ be a fixed positive integer. Find the probability that the particle needs exactly $2n+k$ tosses to reach $(n, n).$

A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once? $ \textbf{(A) } \frac {1}{2187} \qquad \textbf{(B) } \frac {1}{729} \qquad \textbf{(C) } \frac {2}{243} \qquad \textbf{(D) } \frac {1}{81} \qquad \textbf{(E) } \frac {5}{243}$

Tina randomly selects two distinct numbers from the set $ \{1,2,3,4,5\}$ and Sergio randomly selects a number from the set $ \{1,2,...,10\}$. The probability that Sergio's number is larger than the sum of the two numbers chosen by Tina is $ \textbf{(A)}\ 2/5 \qquad \textbf{(B)}\ 9/20 \qquad \textbf{(C)}\ 1/2\qquad \textbf{(D)}\ 11/20 \qquad \textbf{(E)}\ 24/25$

Let $E = \{1, 2, \dots , 16\}$ and let $M$ be the collection of all $4 \times 4$ matrices whose entries are distinct members of $E$. If a matrix $A = (a_{ij} )_{4\times4}$ is chosen randomly from $M$, compute the probability $p(k)$ of $\max_i \min_j a_{ij} = k$ for $k \in E$. Furthermore, determine $l \in E$ such that $p(l) = \max \{p(k) | k \in E \}.$

We choose random a unitary polynomial of degree $n$ and coefficients in the set $1,2,...,n!$. Prove that the probability for this polynomial to be special is between $0.71$ and $0.75$, where a polynomial $g$ is called special if for every $k>1$ in the sequence $f(1), f(2), f(3),...$ there are infinitely many numbers relatively prime with $k$.

A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is probability that the dart lands within the center square? [asy] unitsize(10mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(0,1), B=(1,0), C=(1+sqrt(2),0), D=(2+sqrt(2),1), E=(2+sqrt(2),1+sqrt(2)), F=(1+sqrt(2),2+sqrt(2)), G=(1,2+sqrt(2)), H=(0,1+sqrt(2)); draw(A--B--C--D--E--F--G--H--cycle); draw(A--D); draw(B--G); draw(C--F); draw(E--H); [/asy] $ \textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad\textbf{(E)}\ 2 - \sqrt{2}$

Charles has two six-sided dice. One of the dice is fair, and the other die is biased so that it comes up six with probability $\tfrac23,$ and each of the other five sides has probability $\tfrac{1}{15}.$ Charles chooses one of the two dice at random and rolls it three times. Given that the first two rolls are both sixes, the probability that the third roll will also be a six is $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

A and B play tennis. The player to first win at least four points and at least two more than the other player wins. We know that A gets a point each time with probability $p\le \frac12$, independent of the game so far. Prove that the probability that A wins is at most $2p^2$.

A wishing well is located at the point $(11,11)$ in the $xy$-plane. Rachelle randomly selects an integer $y$ from the set $\left\{ 0, 1, \dots, 10 \right\}$. Then she randomly selects, with replacement, two integers $a,b$ from the set $\left\{ 1,2,\dots,10 \right\}$. The probability the line through $(0,y)$ and $(a,b)$ passes through the well can be expressed as $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. [i]Proposed by Evan Chen[/i]

A palindrome between $ 1000$ and $ 10,000$ is chosen at random. What is the probability that it is divisible by $ 7?$ $ \textbf{(A)}\ \dfrac{1}{10} \qquad \textbf{(B)}\ \dfrac{1}{9} \qquad \textbf{(C)}\ \dfrac{1}{7} \qquad \textbf{(D)}\ \dfrac{1}{6}\qquad \textbf{(E)}\ \dfrac{1}{5}$

Let $ M \equal{} \{ x_1..., x_{30}\}$ a set which consists of 30 distinct positive numbers, let $ A_n,$ $ 1 \leq n \leq 30,$ the sum of all possible products with $ n$ elements each of the set $ M.$ Prove if $ A_{15} > A_{10},$ then $ A_1 > 1.$

Let $P_0P_5Q_5Q_0$ be a rectangular chocolate bar, one half dark chocolate and one half white chocolate, as shown in the diagram below. We randomly select $4$ points on the segment $P_0P_5$, and immediately after selecting those points, we label those $4$ selected points $P_1, P_2, P_3, P_4$ from left to right. Similarly, we randomly select $4$ points on the segment $Q_0Q_5$, and immediately after selecting those points, we label those $4$ points $Q_1, Q_2, Q_3, Q_4$ from left to right. The segments $P_1Q_1, P_2Q_2, P_3Q_3, P_4Q_4$ divide the rectangular chocolate bar into $5$ smaller trapezoidal pieces of chocolate. The probability that exactly $3$ pieces of chocolate contain both dark and white chocolate can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [Diagram in the individuals file for this exam on the Chmmc website]

Richard has a four infinitely large piles of coins: a pile of pennies (worth 1 cent each), a pile of nickels (5 cents), a pile of dimes (10 cents), and a pile of quarters (25 cents). He chooses one pile at random and takes one coin from that pile. Richard then repeats this process until the sum of the values of the coins he has taken is an integer number of dollars. (One dollar is 100 cents.) What is the expected value of this final sum of money, in cents? [i]Proposed by Lewis Chen[/i]

A robot starts at the origin of the Cartesian plane. At each of $10$ steps, he decides to move $ 1$ unit in any of the following directions: left, right, up, or down, each with equal probability. After $10$ steps, the probability that the robot is at the origin is $\frac{n}{4^{10}}$ . Find$ n$

Two jokers are added to a $52$ card deck and the entire stack of $54$ cards is shuffled randomly. What is the expected number of cards that will be strictly between the two jokers?

Consider $ 8$ points that are a knight’s move away from the origin (i.e., the eight points $\{(2, 1)$ , $(2, -1)$ , $(1, 2)$ , $(1, -2)$ , $(-1, 2)$ , $(-1, -2)$ , $(-2, 1)$, $(-2, -1)\}$). Each point has probability $\frac12$ of being visible. What is the expected value of the area of the polygon formed by points that are visible? (If exactly $0, 1, 2$ points appear, this area will be zero.)

Kelvin and Quinn are collecting trading cards; there are $6$ distinct cards that could appear in a pack. Each pack contains exactly one card, and each card is equally likely. Kelvin buys packs until he has at least one copy of every card, and then he stops buying packs. If Quinn is missing exactly one card, the probability that Kelvin has at least two copies of the card Quinn is missing is expressible as $\tfrac{m}{n}$ for coprime positive integers $m$ and $n$. Determine $m+n$.

Coin $ A$ is flipped three times and coin $ B$ is flipped four times. What is the probability that the number of heads obtained from flipping the two fair coins is the same? $ \textbf{(A)}\ \frac {19}{128}\qquad \textbf{(B)}\ \frac {23}{128}\qquad \textbf{(C)}\ \frac {1}{4}\qquad \textbf{(D)}\ \frac {35}{128}\qquad \textbf{(E)}\ \frac {1}{2}$

Expanding $(1+0.2)^{1000}$ by the binomial theorem and doing no further manipulation gives \begin{eqnarray*} &\ & \binom{1000}{0}(0.2)^0+\binom{1000}{1}(0.2)^1+\binom{1000}{2}(0.2)^2+\cdots+\binom{1000}{1000}(0.2)^{1000}\\ &\ & = A_0 + A_1 + A_2 + \cdots + A_{1000}, \end{eqnarray*} where $A_k = \binom{1000}{k}(0.2)^k$ for $k = 0,1,2,\ldots,1000$. For which $k$ is $A_k$ the largest?

You drop a 7 cm long piece of mechanical pencil lead on the floor. A bully takes the lead and breaks it at a random point into two pieces. A piece of lead is unusable if it is 2 cm or shorter. If the expected value of the number of usable pieces afterwards is $\frac{m}n$ for relatively prime positive integers $m$ and $n$, compute $100m + n$. [i]Proposed by Aaron Lin[/i]

A permutation $(a_1, a_2, a_3, \dots, a_{2012})$ of $(1, 2, 3, \dots, 2012)$ is selected at random. If $S$ is the expected value of \[ \sum_{i = 1}^{2012} | a_i - i |, \] then compute the sum of the prime factors of $S$. [i]Proposed by Aaron Lin[/i]

Six men and some number of women stand in a line in random order. Let $p$ be the probability that a group of at least four men stand together in the line, given that every man stands next to at least one other man. Find the least number of women in the line such that $p$ does not exceed 1 percent.

A drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $1/2$ that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with this data?

At a party, $2019$ people decide to form teams of three. To do so, each turn, every person not on a team points to one other person at random. If three people point to each other (that is, $A$ points to $B$, $B$ points to $C$, and $C$ points to $A$), then they form a team. What is the probability that after $65, 536$ turns, exactly one person is not on a team

When flipped, coin A shows heads $\textstyle\frac{1}{3}$ of the time, coin B shows heads $\textstyle\frac{1}{2}$ of the time, and coin C shows heads $\textstyle\frac{2}{3}$ of the time. Anna selects one of the coins at random and flips it four times, yielding three heads and one tail. The probability that Anna flipped coin A can be expressed as $\textstyle\frac{p}{q}$ for relatively prime positive integers $p$ and $q$. Compute $p + q$. [i]Proposed by Eugene Chen[/i]