Dragon selects three positive real numbers with sum $100$, uniformly at random. He asks Cat to copy them down, but Cat gets lazy and rounds them all to the nearest tenth during transcription. If the probability the three new numbers still sum to $100$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, compute $100m+n$. [i]Proposed by Aaron Lin[/i]

Suppose a biased coin gives head with probability $\dfrac{2}{3}$. The coin is tossed repeatedly, if it shows heads then player $A$ rolls a fair die, otherwise player $B$ rolls the same die. The process ends when one of the players get a $6$, and that player is declared the winner. If the probability that $A$ will win is given by $\dfrac{m}{n}$ where $m,n$ are coprime, then what is the value of $m^2n$?

Suppose you have $27$ identical unit cubes colored such that $3$ faces adjacent to a vertex are red and the other $3$ are colored blue. Suppose further that you assemble these $27$ cubes randomly into a larger cube with $3$ cubes to an edge (in particular, the orientation of each cube is random). The probability that the entire cube is one solid color can be written as $\frac{1}{2^n}$ for some positive integer $n$. Find $n$.

Each integer $1$ through $9$ is written on a separate slip of paper and all nine slips are put into a hat. Jack picks one of these slips at random and puts it back. Then Jill picks a slip at random. Which digit is most likely to be the units digit of the [b]sum[/b] of Jack's integer and Jill's integer? $ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ \text{each digit is equally likely} $

An insurance company believes that people can be divided into 2 classes: those who are accident prone and those who are not. Their statistics show that an accident prone person will have an accident in a yearly period with probability 0.4, whereas this probability is 0.2 for the other kind. Given that 30% of people are accident prone, what is the probability that a new policyholder will have an accident within a year of purchasing a policy?

Each of two boxes contains three chips numbered $1$, $2$, $3$. A chip is drawn randomly from each box and the numbers on the two chips are multiplied. What is the probability that their product is even? $\textbf{(A) }\frac{1}{9}\qquad\textbf{(B) }\frac{2}{9}\qquad\textbf{(C) }\frac{4}{9}\qquad\textbf{(D) }\frac{1}{2}\qquad \textbf{(E) }\frac{5}{9}$

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]

We have $2022$ $1s$ written on a board in a line. We randomly choose a strictly increasing sequence from ${1, 2, . . . , 2022}$ such that the last term is $2022$. If the chosen sequence is $a_1, a_2, ..., a_k$ ($k$ is not fixed), then at the $i^{th}$ step, we choose the first a$_i$ numbers on the line and change the 1s to 0s and 0s to 1s. After $k$ steps are over, we calculate the sum of the numbers on the board, say $S$. The expected value of $S$ is $\frac{a}{b}$ where $a, b$ are relatively prime positive integers. Find $a + b.$

The [i]liar's guessing game[/i] is a game played between two players $A$ and $B$. The rules of the game depend on two positive integers $k$ and $n$ which are known to both players. At the start of the game $A$ chooses integers $x$ and $N$ with $1 \le x \le N.$ Player $A$ keeps $x$ secret, and truthfully tells $N$ to player $B$. Player $B$ now tries to obtain information about $x$ by asking player $A$ questions as follows: each question consists of $B$ specifying an arbitrary set $S$ of positive integers (possibly one specified in some previous question), and asking $A$ whether $x$ belongs to $S$. Player $B$ may ask as many questions as he wishes. After each question, player $A$ must immediately answer it with [i]yes[/i] or [i]no[/i], but is allowed to lie as many times as she wants; the only restriction is that, among any $k+1$ consecutive answers, at least one answer must be truthful. After $B$ has asked as many questions as he wants, he must specify a set $X$ of at most $n$ positive integers. If $x$ belongs to $X$, then $B$ wins; otherwise, he loses. Prove that: 1. If $n \ge 2^k,$ then $B$ can guarantee a win. 2. For all sufficiently large $k$, there exists an integer $n \ge (1.99)^k$ such that $B$ cannot guarantee a win. [i]Proposed by David Arthur, Canada[/i]

The train schedule in Hummut is hopelessly unreliable. Train $A$ will enter Intersection $X$ from the west at a random time between $9:00$ am and $2:30$ pm; each moment in that interval is equally likely. Train $B$ will enter the same intersection from the north at a random time between $9:30$ am and $12:30$ pm, independent of Train $A$; again, each moment in the interval is equally likely. If each train takes $45$ minutes to clear the intersection, what is the probability of a collision today?

Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible? $ \textbf{(A)}\ 2520 \qquad \textbf{(B)}\ 2880 \qquad \textbf{(C)}\ 3120 \qquad \textbf{(D)}\ 3250 \qquad \textbf{(E)}\ 3750 $

Evaluate the limit: \[ \lim_{n \to \infty}{n \cdot \sin{1} \cdot \sin{2} \cdot \dots \cdot \sin{n}}.\] Proposed to "Unirea" Intercounty contest, grade 11, Romania

Let $ \xi_1,\xi_2,...$ be independent, identically distributed random variables with distribution \[ P(\xi_1=-1)=P(\xi_1=1)=\frac 12 .\] Write $ S_n=\xi_1+\xi_2+...+\xi_n \;(n=1,2,...),\ \;S_0=0\ ,$ and \[ T_n= \frac{1}{\sqrt{n}} \max _{ 0 \leq k \leq n}S_k .\] Prove that $ \liminf_{n \rightarrow \infty} (\log n)T_n=0$ with probability one. [i]P. Revesz[/i]

Aileen plays badminton where she and her opponent stand on opposite sides of a net and attempt to bat a birdie back and forth over the net. A player wins a point if their opponent fails to bat the birdie over the net. When Aileen is the server (the first player to try to hit the birdie over the net), she wins a point with probability $\frac{9}{10}$ . Each time Aileen successfully bats the birdie over the net, her opponent, independent of all previous hits, returns the birdie with probability $\frac{3}{4}$ . Each time Aileen bats the birdie, independent of all previous hits, she returns the birdie with probability $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let $\mathbb X$ be the set of all bijective functions from the set $S=\{1,2,\cdots, n\}$ to itself. For each $f\in \mathbb X,$ define \[T_f(j)=\left\{\begin{aligned} 1, \ \ \ & \text{if} \ \ f^{(12)}(j)=j,\\ 0, \ \ \ & \text{otherwise}\end{aligned}\right.\] Determine $\sum_{f\in\mathbb X}\sum_{j=1}^nT_{f}(j).$ (Here $f^{(k)}(x)=f(f^{(k-1)}(x))$ for all $k\geq 2.$)

Let $ S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $ P$ whose vertices are in $ S$, let $ a(P)$ be the number of vertices of $ P$, and let $ b(P)$ be the number of points of $ S$ which are outside $ P$. A line segment, a point, and the empty set are considered as convex polygons of $ 2$, $ 1$, and $ 0$ vertices respectively. Prove that for every real number $ x$ \[\sum_{P}{x^{a(P)}(1 \minus{} x)^{b(P)}} \equal{} 1,\] where the sum is taken over all convex polygons with vertices in $ S$. [i]Alternative formulation[/i]: Let $ M$ be a finite point set in the plane and no three points are collinear. A subset $ A$ of $ M$ will be called round if its elements is the set of vertices of a convex $ A \minus{}$gon $ V(A).$ For each round subset let $ r(A)$ be the number of points from $ M$ which are exterior from the convex $ A \minus{}$gon $ V(A).$ Subsets with $ 0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $ A$ of $ M$ construct the polynomial \[ P_A(x) \equal{} x^{|A|}(1 \minus{} x)^{r(A)}. \] Show that the sum of polynomials for all round subsets is exactly the polynomial $ P(x) \equal{} 1.$ [i]Proposed by Federico Ardila, Colombia[/i]

Let $\mathcal{N}_p$ stand for a $p$ dimensional random variable of standard normal distribution. For $a\in\mathbb{R}^p$, let $H_p(a)$ stand for the expectation $E|\mathcal{N}_p+a|$. For $p>1$, prove that $$H_p(a)=(p-1)\int_0^{\infty} H_1\left( \frac{|a|}{\sqrt{r^2+1}}\right) \frac{r^{p-2}}{\sqrt{(r^2+1)^p}} \mathrm{d}r$$

Let $n\geq 3$ be integer. Given two pairs of $n$ cards numbered from 1 to $n$. Mix the $2n$ cards up and take the card 3 times every one card. Denote $X_1,\ X_2,\ X_3$ the numbers of the cards taken out in this order taken the cards. Find the probabilty such that $X_1<X_2<X_3$. Note that once a card taken out, it is not taken a back.

A game in a casino uses the diagram shown. At the start a ball appears at $S$. Each time the player presses a button, the ball moves to one of the adjacent letters with equal probability. The game ends when one of the following two things happens: (i) The ball returns to $S$, the player loses. (ii) The ball reaches $G$, the player wins. Find the probability that the player wins and the expected duration of a game.

Raashan, Sylvia, and Ted play the following game. Each starts with $\$1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $\$1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have $\$1$? (For example, Raashan and Ted may each decide to give $\$1$ to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $\$0$, Sylvia would have $\$2$, and Ted would have $\$1$, and and that is the end of the first round of play. In the second round Raashan has no money to give, but Sylvia and Ted might choose each other to give their $\$1$ to, and and the holdings will be the same as the end of the second [sic] round. $\textbf{(A) } \frac{1}{7} \qquad\textbf{(B) } \frac{1}{4} \qquad\textbf{(C) } \frac{1}{3} \qquad\textbf{(D) } \frac{1}{2} \qquad\textbf{(E) } \frac{2}{3}$

A point $(X, Y, Z)$ is chosen uniformly at random from the ball of radius $4$ centered at the origin (i.e., the set $\{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 \leq 4^2\}$). Compute the probability that the inequalities $X^2 \leq 1$ and $X^2 + Y^2 + Z^2 \geq 1$ simultaneously hold.

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}$

Three cards, only one of which is an ace, are placed face down on a table. You select one, but do not look at it. The dealer turns over one of the other cards, which is not the ace (if neither are, he picks one of them randomly to turn over). You get a chance to change your choice and pick either of the remaining two face-down cards. If you selected the cards so as to maximize the chance of finding the ace on the second try, what is the probability that you selected it on the    (a) first try?    (b) second try?

A fair coin is tossed repeatedly until there is a run of an odd number of heads followed by a tail. Determine the expected number of tosses.

Given a random string of 33 bits (0 or 1), how many (they can overlap) occurrences of two consecutive 0's would you expect? (i.e. "100101" has 1 occurrence, "0001" has 2 occurrences)