Emily starts with an empty bucket. Every second, she either adds a stone to the bucket or removes a stone from the bucket, each with probability $\frac{1}{2}$. If she wants to remove a stone from the bucket and the bucket is currently empty, she merely does nothing for that second (still with probability $\hfill \frac{1}{2}$). What is the probability that after $2017$ seconds her bucket contains exactly $1337$ stones?

$2019$ points are chosen at random, independently, and distributed uniformly in the unit disc $\{(x,y)\in\mathbb R^2: x^2+y^2\le 1\}$. Let $C$ be the convex hull of the chosen points. Which probability is larger: that $C$ is a polygon with three vertices, or a polygon with four vertices? [i]Proposed by Fedor Petrov, St. Petersburg State University[/i]

Each citizen in a town knows at least $ 30$% of the remaining citizens. A citizen votes in elections if he/she knows at least one candidate. Prove that it is possible to schedule elections with two candidates for the mayor of the city so that at least $ 50$% of the citizen can vote.

When 7 fair standard 6-sided dice are thrown, the probability that the sum of the numbers on the top faces is 10 can be written as $$\frac{n}{6^7},$$where $n$ is a positive integer. What is $n$? $\textbf{(A) } 42 \qquad \textbf{(B) } 49 \qquad \textbf{(C) } 56 \qquad \textbf{(D) } 63 \qquad \textbf{(E) } 84 $

There are seven cards in a hat, and on the card $k$ there is a number $2^{k-1}$, $k=1,2,...,7$. Solarin picks the cards up at random from the hat, one card at a time, until the sum of the numbers on cards in his hand exceeds $124$. What is the most probable sum he can get?

Three people, John, Macky, and Rik, play a game of passing a basketball from one to another. Find the number of ways of passing the ball starting with Macky and reaching Macky again at the end of the seventh pass.

Suppose $x$ is a random real number between $1$ and $4$, and $y$ is a random real number between $1$ and $9$. If the expected value of \[ \left\lceil \log_2 x \right\rceil - \left\lfloor \log_3 y \right\rfloor \] can be expressed as $\frac mn$ where $m$ and $n$ are relatively prime positive integers, compute $100m + n$. [i]Proposed by Lewis Chen[/i]

Each spinner is divided into $3$ equal parts. The results obtained from spinning the two spinners are multiplied. What is the probability that this product is an even number? [asy] draw(circle((0,0),2)); draw(circle((5,0),2)); draw((0,0)--(sqrt(3),1)); draw((0,0)--(-sqrt(3),1)); draw((0,0)--(0,-2)); draw((5,0)--(5+sqrt(3),1)); draw((5,0)--(5-sqrt(3),1)); draw((5,0)--(5,-2)); fill((0,5/3)--(2/3,7/3)--(1/3,7/3)--(1/3,3)--(-1/3,3)--(-1/3,7/3)--(-2/3,7/3)--cycle,black); fill((5,5/3)--(17/3,7/3)--(16/3,7/3)--(16/3,3)--(14/3,3)--(14/3,7/3)--(13/3,7/3)--cycle,black); label("$1$",(0,1/2),N); label("$2$",(sqrt(3)/4,-1/4),ESE); label("$3$",(-sqrt(3)/4,-1/4),WSW); label("$4$",(5,1/2),N); label("$5$",(5+sqrt(3)/4,-1/4),ESE); label("$6$",(5-sqrt(3)/4,-1/4),WSW); [/asy] $\text{(A)}\ \frac{1}{3} \qquad \text{(B)}\ \frac{1}{2} \qquad \text{(C)}\ \frac{2}{3} \qquad \text{(D)}\ \frac{7}{9} \qquad \text{(E)}\ 1$

A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point $ P$ is selected at random inside the circumscribed sphere. The probability that $ P$ lies inside one of the five small spheres is closest to $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 0.1\qquad \textbf{(C)}\ 0.2\qquad \textbf{(D)}\ 0.3\qquad \textbf{(E)}\ 0.4$

Let a sequence be defined as follows: $a_0=1$, and for $n>0$, $a_n$ is $\tfrac{1}{3}a_{n-1}$ and is $\tfrac{1}{9}a_{n-1}$ with probability $\tfrac{1}{2}$. If the expected value of $\textstyle\sum_{n=0}^{\infty}a_n$ can be expressed in simplest form as $\tfrac{p}{q}$, what is $p+q$?

Let $0<p<1$ be given. Initially, we have $n$ coins, all of which have probability $p$ of landing on heads, and probability $1-p$ of landing on tails (the results of the tosses are independent of each other). In each round, we toss our coins and remove those that result in heads. We keep repeating this until all our coins are removed. Let $k_n$ denote the expected number of rounds that are needed to get rid of all the coins. Prove that there exists $c>0$ for which the following inequality holds for all $n>0$ \[c\bigg(1+\frac{1}{2}+\cdots+\frac{1}{n}\bigg)<k_n<1+c\bigg(1+\frac{1}{2}+\cdots+\frac{1}{n}\bigg).\]

Diana and Apollo each roll a standard die obtaining a number at random from $1$ to $6$. What is the probability that Diana's number is larger than Apollo's number? $\text{(A)}\ \dfrac{1}{3} \qquad \text{(B)}\ \dfrac{5}{12} \qquad \text{(C)}\ \dfrac{4}{9} \qquad \text{(D)}\ \dfrac{17}{36} \qquad \text{(E)}\ \dfrac{1}{2}$

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]

Ankit, Box, and Clark are taking the tiebreakers for the geometry round, consisting of three problems. Problem $k$ takes each $k$ minutes to solve. If for any given problem there is a $\frac13$ chance for each contestant to solve that problem first, what is the probability that Ankit solves a problem first?

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$.)

Pick a $3$-digit number $abc$, which contains no $0$'s. The probability that this is a winning number is $\frac1a\cdot\frac1b\cdot\frac1c$. However, the BMT problem writer tries to balance out the chances for the numbers in the following ways: [list] [*] For the lowest digit $n$ in the number, he rolls an $n$-sided die for each time that the digit appears, and gives the number $0$ probability of winning if an $n$ is rolled. [*] For the largest digit $m$ in the number, he rolls an $m$-sided die once and scales the probability of winning by that die roll. [/list] If you choose optimally, what is the probability that your number is a winning number?

Let $(\Omega, \mathcal A, P)$ be a probability space, and let $(X_n, \mathcal F_n)$ be an adapted sequence in $(\Omega, \mathcal A, P)$ (that is, for the $\sigma$-algebras $\mathcal F_n$, we have $\mathcal F_1\subseteq \mathcal F_2\subseteq \dots \subseteq \mathcal A$, and for all $n$, $X_n$ is an $\mathcal F_n$-measurable and integrable random variable). Assume that $$\mathrm E (X_{n+1} \mid \mathcal F_n )=\frac12 X_n+\frac12 X_{n-1}\,\,\,\,\, (n=2, 3, \ldots )$$ Prove that $\mathrm{sup}_n \mathrm{E}|X_n|<\infty$ implies that $X_n$ converges with probability one as $n\to\infty$. [I. Fazekas]

Compute the expected sum of elements in a subset of $\{1, 2, 3, . . . , 2020\}$ (including the empty set) chosen uniformly at random.

Let $a/b$ be the probability that a randomly selected divisor of $2007$ is a multiple of $3$. If $a$ and $b$ are relatively prime positive integers, find $a+b$.

Aws plays a solitaire game on a fifty-two card deck: whenever two cards of the same color are adjacent, he can remove them. Aws wins the game if he removes all the cards. If Aws starts with the cards in a random order, what is the probability for him to win?

One student in a class of boys and girls is chosen to represent the class. Each student is equally likely to be chosen and the probability that a boy is chosen is $ \frac23$ of the probability that a girl is chosen. The ratio of the number of boys to the total number of boys and girls is $ \textbf{(A)}\ \frac13 \qquad \textbf{(B)}\ \frac25 \qquad \textbf{(C)}\ \frac12 \qquad \textbf{(D)}\ \frac35 \qquad \textbf{(E)}\ \frac23$

Alex starts at the origin $O$ of a hexagonal lattice. Every second, he moves to one of the six vertices adjacent to the vertex he is currently at. If he ends up at $X$ after $2018$ moves, then let $p$ be the probability that the shortest walk from $O$ to $X$ (where a valid move is from a vertex to an adjacent vertex) has length $2018$. Then $p$ can be expressed as $\tfrac{a^m-b}{c^n}$, where $a$, $b$, and $c$ are positive integers less than $10$; $a$ and $c$ are not perfect squares; and $m$ and $n$ are positive integers less than $10000$. Find $a+b+c+m+n$.

$N$ unfair coins (with heads and tails on the sides) are thrown, with the $k^{th}$ coin has got a chance of $\frac{1}{2k + 1}$ to land on tails.How high is the probability that an odd number of coins will show tails?

Investigate the behaviour at $t\to\infty$ of the solution of the problem \[u_t+(u\sin x)_x=\epsilon u_{xx},\;u|_{t=0}=1,\;\epsilon\ll1\]

Darryl has a six-sided die with faces $1, 2, 3, 4, 5, 6$. He knows the die is weighted so that one face comes up with probability $1/2$ and the other five faces have equal probability of coming up. He unfortunately does not know which side is weighted, but he knows each face is equally likely to be the weighted one. He rolls the die $5$ times and gets a $1, 2, 3, 4$ and $5$ in some unspecified order. Compute the probability that his next roll is a $6$.