My frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score. The game ends after one of the two teams scores three points (total, not necessarily consecutive). If every possible sequence of scores is equally likely, what is the expected score of the losing team? $\textbf{(A) }2/3\hspace{14em}\textbf{(B) }1\hspace{14.8em}\textbf{(C) }3/2$ $\textbf{(D) }8/5\hspace{14em}\textbf{(E) }5/8\hspace{14em}\textbf{(F) }2$ $\textbf{(G) }0\hspace{14.9em}\textbf{(H) }5/2\hspace{14em}\textbf{(I) }2/5$ $\textbf{(J) }3/4\hspace{14em}\,\textbf{(K) }4/3\hspace{13.9em}\textbf{(L) }2007$

Call a positive rational number in simplest terms [i]coddly[/i] if its numerator and denominator are both odd. Consider the equation $$2017= x_1\text{ }\square\text{ }x_2\text{ }\square\text{ }x_3\text{ }\ldots \text{ }\square \text{ }x_{2016} \text{ }\square \text{ }x_{2017},$$ where there are $2016$ boxes. We fill up the boxes randomly with the operations $+$, $-$, and $\times$. Compute the probability that there exists a solution in [b]distinct[/b] coddly numbers $(x_1,x_2, \ldots x_{2017})$ to the resulting equation.

Let $S$ be the set of positive integer divisors of $20^9.$ Three numbers are chosen independently and at random from the set $S$ and labeled $a_1,a_2,$ and $a_3$ in the order they are chosen. The probability that both $a_1$ divides $a_2$ and $a_2$ divides $a_3$ is $\frac mn,$ where $m$ and $n$ are relatively prime positive integers. Find $m.$

An $m\times n$ checkerboard is colored randomly: each square is independently assigned red or black with probability $\frac12.$ we say that two squares, $p$ and $q$, are in the same connected monochromatic region if there is a sequence of squares, all of the same color, starting at $p$ and ending at $q,$ in which successive squares in the sequence share a common side. Show that the expected number of connected monochromatic regions is greater than $\frac{mn}8.$

Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0,1]$. Two random numbers $x$ and $y$ are chosen independently in this manner. What is the probability that $|x-y| > \tfrac{1}{2}$? $\textbf{(A)} \frac{1}{3} \qquad \textbf{(B)} \frac{7}{16} \qquad \textbf{(C)} \frac{1}{2} \qquad \textbf{(D)} \frac{9}{16} \qquad \textbf{(E)} \frac{2}{3}$

John is flipping his favorite bottle, which currently contains $10$ ounces of water. However, his bottle is broken from excessive flipping, so after he performs a flip, one ounce of water leaks out of his bottle. When his bottle contains k ounces of water, he has a $\frac{1}{k+1}$ probability of landing it on its bottom. What is the expected number of number of flips it takes for John’s bottle to land on its bottom ?

A radio program has a quiz consisting of $3$ multiple-choice questions, each with $3$ choices. A contestant wins if he or she gets $2$ or more of the questions right. The contestant answers randomly to each question. What is the probability of winning? $\textbf{(A) } \frac{1}{27}\qquad \textbf{(B) } \frac{1}{9}\qquad \textbf{(C) } \frac{2}{9}\qquad \textbf{(D) } \frac{7}{27}\qquad \textbf{(E) } \frac{1}{2}$

Two persons, $X$ and $Y$ , play with a die. $X$ wins a game if the outcome is $1$ or $2$; $Y$ wins in the other cases. A player wins a match if he wins two consecutive games. For each player determine the probability of winning a match within $5$ games. Determine the probabilities of winning in an unlimited number of games. If $X$ bets $1$, how much must $Y$ bet for the game to be fair ?

A game consists in pushing a flat stone along a sequence of squares $S_0, S_1, S_2, . . .$ that are arranged in linear order. The stone is initially placed on square $S_0$. When the stone stops on a square $S_k$ it is pushed again in the same direction and so on until it reaches $S_{1987}$ or goes beyond it; then the game stops. Each time the stone is pushed, the probability that it will advance exactly $n$ squares is $\frac{1}{2^n}$. Determine the probability that the stone will stop exactly on square $S_{1987}.$

Joe and Kathryn work part-time jobs at the local mall to make some money for college. Joe works at GameStop, while Kathryn works at Bath and Body Works. However, neither of them usually ever leaves on pay day without spending a healthy portion of their check at their own store, especially angering Joe’s parents, who think video games are for Neanderthals or children under $8$. Joe makes $\$8$ an hour, while Kathryn makes $\$10$ an hour. Both work $20$ hours a week. Every week, Joe has a $20\%$ probability of purchasing a used $\$25$ video game, and Kathryn has a $25\%$ probability of purchasing a $\$30$ skin moisturizer. Find the expected value, in dollars, of their combined weekly “take-home pay.” (Take-home pay is total pay minus in-store spending.)

Natalie has a copy of the unit interval $[0,1]$ that is colored white. She also has a black marker, and she colors the interval in the following manner: at each step, she selects a value $x\in [0,1]$ uniformly at random, and (a) If $x\leq\tfrac12$ she colors the interval $[x,x+\tfrac12]$ with her marker. (b) If $x>\tfrac12$ she colors the intervals $[x,1]$ and $[0,x-\tfrac12]$ with her marker. What is the expected value of the number of steps Natalie will need to color the entire interval black?

Now Wendy wanders over and joins Dr. Lisi and her younger siblings. Thinking she knows everything there is about how to work with arithmetic series, she nearly turns right around to walk back home when Dr. Lisi poses a more challenging problem. "Suppose I select two distinct terms at random from the $2008$ term sequence. What's the probability that their product is positive?" If $a$ and $b$ are relatively prime positive integers such that $a/b$ is the probability that the product of the two terms is positive, find the value of $a+b$.

[b]9.[/b] Let $X_0, X_1, \dots $ be independent, indentically distributed, nondegenerate random variables, and let $0<\alpha <1$ be a real number. Assume that the series $\sum_{k=1}^{\infty} \alpha^{k} X_k$ is convergent with probability one. Prove that the distribution function of the sum is continuous. ([b]P. 23[/b]) [T. F. Móri]

Wendy randomly chooses a positive integer less than or equal to $2020$. The probability that the digits in Wendy's number add up to $10$ is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Mikko takes a multiple choice test with ten questions. His only goal is to pass the test, and this requires seven points. A correct answer is worth one point, and answering wrong results in the deduction of one point. Mikko knows for sure that he knows the correct answer in the six first questions. For the rest, he estimates that he can give the correct answer to each problem with probability $p, 0 < p < 1$. How many questions Mikko should try?

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.

Leo and Paul are at the Berkeley BART station and are racing to San Francisco. Leo is planning to take the line that takes him directly to SF, and because he has terrible BART luck, his train will arrive in some integer number of minutes, with probability $\frac i{210}$ for $1\le i\le20$ at any given minute. Paul will take a second line, whose trains always arrive before Leo’s train, with uniform probability. However, Paul must also make a transfer to a 3rd line, whose trains arrive with uniform probability between $0$ and $10$ minutes after Paul reaches the transfer station. What is the probability that Leo gets to SF before Paul does?

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]

Ahuiliztli is playing around with some coins (pennies, nickels, dimes, and quarters). She keeps grabbing $k$ coins and calculating the value of her handful. After a while, she begins to notice that if $k$ is even, she more often gets even sums, and if $k$ is odd, she more often gets odd sums. Help her prove this true! Given $k$ coins chosen uniformly and at random, prove that. the probability that the parity of $k$ is the same as the parity of the $k$ coins' value is greater than the probability that the parities are different.

Seven people of seven different ages are attending a meeting. The seven people leave the meeting one at a time in random order. Given that the youngest person leaves the meeting sometime before the oldest person leaves the meeting, the probability that the third, fourth, and fifth people to leave the meeting do so in order of their ages (youngest to oldest) is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n$.

Let there be $n$ students, numbered $1$ through $n$. Let there be $n$ cards with numbers $1$ through $n$ written on them. Each student picks a card from the stack, and two students are called a pair if they pick each other's number. Let the probability that there are no pairs be $p_n$. Prove that $p_n - p_{n-1}=0$ if $n$ is odd, and prove that $p_n - p_{n-1}= \frac{1}{(-2)^kk^{1-k}}$ if $n = 2k$.

The four faces of a tetrahedral die are labelled $0, 1, 2,$ and $3,$ and the die has the property that, when it is rolled, the die promptly vanishes, and a number of copies of itself appear equal to the number on the face the die landed on. For example, if it lands on the face labelled $0,$ it disappears. If it lands on the face labelled $1,$ nothing happens. If it lands on the face labelled $2$ or $3,$ there will then be $2$ or $3$ copies of the die, respectively (including the original). Suppose the die and all its copies are continually rolled, and let $p$ be the probability that they will all eventually disappear. Find $\left\lfloor \frac{10}{p} \right\rfloor$.

An unfair coin lands heads with probability $\tfrac1{17}$ and tails with probability $\tfrac{16}{17}$. Matt flips the coin repeatedly until he flips at least one head and at least one tail. What is the expected number of times that Matt flips the coin?

Actually, this is an MO I participated in :) but it's really hard to get problems from this year if you don't know some people. P1. Let $f$ be a function satisfying $f(x + 1) + f(x - 1) = \sqrt{2} f(x)$, for all reals $x$. If $f(x - 1) = a$ and $f(x) = b$, determine the value of $f(x + 4)$. [hide=Remarks]We found out that this is the modified version of a problem from LMNAS UGM 2008, Senior High School Level, on its First Round. This is also the same with Arthur Engel's "Problem Solving Strategies" Book, Example Problem E2.[/hide] P2. The sequence of "Sanga" numbers is formed by the following procedure. i. Pick a positive integer $n$. ii. The first term of the sequence $(U_1)$ is $9n$. iii. For $k \geq 2$, $U_k = U_{k-1} - 17$. Sanga$[r]$ is the "Sanga" sequence whose smallest positive term is $r$. As an example, for $n = 3$, the "Sanga" sequence which is formed is $27, 10, -7, -24, -41, \ldots.$ Since the smallest positive term of such sequence is $10$, for $n = 3$, the sequence formed is called Sanga$[10]$. For $n \leq 100$, determine the sum of all $n$ which makes the sequence Sanga$[4]$. P3. The cube $ABCD.EFGH$ has an edge length of 6 cm. Point $R$ is on the extension of line (segment) $EH$ with $EH : ER = 1 : 2$, such that triangle $AFR$ cuts edge $GH$ at point $P$ and cuts edge $DH$ at $Q$. Determine the area of the region bounded by the quadrilateral $AFPQ$. [url=https://artofproblemsolving.com/community/q1h2395046p19649729]P4[/url]. Ten skydivers are planning to form a circle formation when they are in the air by holding hands with both adjacent skydivers. If each person has 2 choices for the colour of his/her uniform to be worn, that is, red or white, determine the number of different colour formations that can be constructed. P5. After pressing the start button, a game machine works according to the following procedure. i. It picks 7 numbers randomly from 1 to 9 (these numbers are integers, not stated but corrected) without showing it on screen. ii. It shows the product of the seven chosen numbes on screen. iii. It shows a calculator menu (it does not function as a calculator) on screen and asks the player whether the sum of the seven chosen numbers is odd or even. iv. Shows the seven chosen numbers and their sum and products. v. Releases a prize if the guess of the player was correct or shows the message "Try again" on screen if the guess by the player was incorrect. (Although the player is not allowed to guess with those numbers, and the machine's procedures are started all over again.) Kiki says that this game is really easy since the probability of winning is greater than $90$%. Explain, whether you agree with Kiki.

Let $a$ and $b$ be given positive real numbers, with $a<b.$ If two points are selected at random from a straight line segment of length $b,$ what is the probability that the distance between them is at least $a?$