Marko lives on the origin of the Cartesian plane. Every second, Marko moves $1$ unit up with probability $\tfrac{2}{9}$, $1$ unit right with probability $\tfrac{2}{9}$, $1$ unit up and $1$ unit right with probability $\tfrac{4}{9}$, and he doesn’t move with probability $\tfrac{1}{9}$. After $2019$ seconds, Marko ends up on the point $(A, B)$. What is the expected value of $A\cdot B$?

For each $i=1,2,..N$, let $a_i,b_i,c_i$ be integers such that at least one of them is odd. Show that one can find integers $x,y,z$ such that $xa_i+yb_i+zc_i$ is odd for at least $\frac{4}{7}N$ different values of $i$.

Tim is participating in the following three math contests. On each contest his score is the number of correct answers. $\bullet$ The Local Area Inspirational Math Exam consists of 15 problems. $\bullet$ The Further Away Regional Math League has 10 problems. $\bullet$ The Distance-Optimized Math Open has 50 problems. For every positive integer $n$, Tim knows the answer to the $n$th problems on each contest (which are pairwise distinct), if they exist; however, these answers have been randomly permuted so that he does not know which answer corresponds to which contest. Unaware of the shuffling, he competes with his modified answers. Compute the expected value of the sum of his scores on all three contests. [i]Proposed by Evan Chen[/i]

Two fair octahedral dice, each with the numbers $1$ through $8$ on their faces, are rolled. Let $N$ be the remainder when the product of the numbers showing on the two dice is divided by $8$. Find the expected value of $N$.

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 $p+q$?

Yannick is playing a game with $100$ rounds, starting with $1$ coin. During each round, there is an $n\%$ chance that he gains an extra coin, where $n$ is the number of coins he has at the beginning of the round. What is the expected number of coins he will have at the end of the game?

Let $a_1,a_2,a_3,\ldots$ be an infinite sequence where for all positive integers $i$, $a_i$ is chosen to be a random positive integer between $1$ and $2016$, inclusive. Let $S$ be the set of all positive integers $k$ such that for all positive integers $j<k$, $a_j\neq a_k$. (So $1\in S$; $2\in S$ if and only if $a_1\neq a_2$; $3\in S$ if and only if $a_1\neq a_3$ and $a_2\neq a_3$; and so on.) In simplest form, let $\dfrac{p}{q}$ be the expected number of positive integers $m$ such that $m$ and $m+1$ are in $S$. Compute $pq$.

Bob has invented the Very Normal Coin (VNC). When the VNC is flipped, it shows heads $\textstyle\frac{1}{2}$ of the time and tails $\textstyle\frac{1}{2}$ of the time - unless it has yielded the same result five times in a row, in which case it is guaranteed to yield the opposite result. For example, if Bob flips five heads in a row, then the next flip is guaranteed to be tails. Bob flips the VNC an infinite number of times. On the $n$th flip, Bob bets $2^{-n}$ dollars that the VNC will show heads (so if the second flip shows heads, Bob wins $\$0.25$, and if the third flip shows tails, Bob loses $\$0.125$). Assume that dollars are infinitely divisible. Given that the first flip is heads, the expected number of dollars Bob is expected to win can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$. [i]Proposed by Lewis Chen[/i]

Let $a_1,a_2,...,a_9$ be nine real numbers, not necessarily distinct, with average $m$. Let $A$ denote the number of triples $1 \le i < j < k \le 9$ for which $a_i + a_j + a_k \ge 3m$. What is the minimum possible value of $A$? [i]Proposed by Ray Li[/i]

A tree $T$ is given. Starting with the complete graph on $n$ vertices, subgraphs isomorphic to $T$ are erased at random until no such subgraph remains. For what trees does there exist a positive constant $c$ such that the expected number of edges remaining is at least $cn^2$ for all positive integers $n$? [i]David Yang.[/i]

There are 94 safes and 94 keys. Each key can open only one safe, and each safe can be opened by only one key. We place randomly one key into each safe. 92 safes are then randomly chosen, and then locked. What is the probability that we can open all the safes with the two keys in the two remaining safes? (Once a safe is opened, the key inside the safe can be used to open another safe.)

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 $n>0$ \[c\bigg(1+\frac{1}{2}+\cdots+\frac{1}{n}\bigg)<k_n<1+c\bigg(1+\frac{1}{2}+\cdots+\frac{1}{n}\bigg).\]

Let $B$ and $D$ be two points chosen independently and uniformly at random from the unit sphere in 3D space centered at a point $A$ (this unit sphere is the set of all points in $\mathbb{R}^3$ a distance of $1$ away from $A$). Compute the expected value of $\sin^2\angle DAB$.

Concentric circles $\Omega_1$ and $\Omega_2$ with radii $1$ and $100$, respectively, are drawn with center $O$. Points $A$ and $B$ are chosen independently at random on the circumferences of $\Omega_1$ and $\Omega_2$, respectively. Denote by $\ell$ the tangent line to $\Omega_1$ passing through $A$, and denote by $P$ the reflection of $B$ across $\ell$. Compute the expected value of $OP^2$. [i]Proposed by Lewis Chen[/i]

A rock is dropped off a cliff of height $ h $ As it falls, a camera takes several photographs, at random intervals. At each picture, I measure the distance the rock has fallen. Let the average (expected value) of all of these distances be $ kh $. If the number of photographs taken is huge, find $ k $. That is: what is the time-average of the distance traveled divided by $ h $, dividing by $h$? $ \textbf {(A) } \dfrac{1}{4} \qquad \textbf {(B) } \dfrac{1}{3} \qquad \textbf {(C) } \dfrac{1}{\sqrt{2}} \qquad \textbf {(D) } \dfrac{1}{2} \qquad \textbf {(E) } \dfrac{1}{\sqrt{3}} $ [i]Problem proposed by Ahaan Rungta[/i]

The average of all ten-digit base-ten positive integers $\underline{d_9} \ \underline{d_8} \ldots \underline{d_1} \ \underline{d_0}$ that satisfy the property $|d_i - i| \leq 1$ for all $i \in \{0, 1, \ldots, 9\}$ is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime integers. Compute the remainder when $p + q$ is divided by $10^6.$

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 randomly selected four-element subset of $\{1, 2, 3, 4, 5, 6, 7, 8\}$. Let $m$ and $n$ be relatively prime positive integers so that the expected value of the maximum element in $S$ is $\dfrac{m}{n}$. 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.

Let $N{}$ be a fixed positive integer, $S{}$ be the set $\{1, 2,\ldots , N\}$ and $\mathcal{F}$ be the set of functions $f:S\to S$ such that $f(i)\geqslant i$ for all $i\in S.$ For each $f\in\mathcal{F}$ let $P_f$ be the unique polynomial of degree less than $N{}$ satisfying $P_f(i) = f(i)$ for all $i\in S.$ If $f{}$ is chosen uniformly at random from $\mathcal{F}$ determine the expected value of $P_f'(0)$ where\[P_f'(0)=\frac{\mathrm{d}P_f(x)}{\mathrm{d}x}\bigg\vert_{x=0}.\][i]Proposed by Ishan Nath[/i]

At first a fair dice is placed in such way the spot $1$ is shown on the top face. After that, select a face from the four sides at random, the process that the side would be the top face is repeated $n$ times. Note the sum of the spots of opposite face is 7. (1) Find the probability such that the spot $1$ would appear on the top face after the $n$-repetition. (2) Find the expected value of the number of the spot on the top face after the $n$-repetition.

Sophie has $20$ indistinguishable pairs of socks in a laundry bag. She pulls them out one at a time. After pulling out $30$ socks, the expected number of unmatched socks among the socks that she has pulled out can be expressed in simplest form as $\tfrac{m}{n}$. Find $m+n$.

Kevin has $255$ cookies, each labeled with a unique nonempty subset of $\{1,2,3,4,5,6,7,8\}$. Each day, he chooses one cookie uniformly at random out of the cookies not yet eaten. Then, he eats that cookie, and all remaining cookies that are labeled with a subset of that cookie (for example, if he chooses the cookie labeled with $\{1,2\}$, he eats that cookie as well as the cookies with $\{1\}$ and $\{2\}$). The expected value of the number of days that Kevin eats a cookie before all cookies are gone can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [i]Proposed by Ray Li[/i]

assume that we have a n*n table we fill it with 1,...,n such that each number exists exactly n times prove that there exist a row or column such that at least $\sqrt{n}$ diffrent number are contained.

Dr. G has a bag of five marbles and enjoys drawing one marble from the bag, uniformly at random, and then putting it back in the bag. How many draws, on average, will it take Dr. G to reach a point where every marble has been drawn at least once?