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]

Let $n$ be a positive integer. Let $f(n)$ be the probability that, if divisors $a, b, c$ of $n$ are selected uniformly at random with replacement, then $\gcd(a, \text{lcm}(b, c)) = \text{lcm}(a, \gcd(b, c))$. Let $s(n)$ be the sum of the distinct prime divisors of $n$. If $f(n) < \frac{1}{2018}$, compute the smallest possible value of $s(n)$.

Four children were born at City Hospital yesterday. Assume each child is equally likely to be a boy or a girl. Which of the following outcomes is most likely? $ \textbf{(A) }\text{all 4 are boys}$\\ $\textbf{(B) }\text{all 4 are girls}$\\$ \textbf{(C) }\text{2 are girls and 2 are boys}$\\ $\textbf{(D) }\text{3 are of one gender and 1 is of the other gender}$\\ $\textbf{(E) }\text{all of these outcomes are equally likely} $

A Sudoku matrix is defined as a $ 9\times9$ array with entries from $ \{1, 2, \ldots , 9\}$ and with the constraint that each row, each column, and each of the nine $ 3 \times 3$ boxes that tile the array contains each digit from $ 1$ to $ 9$ exactly once. A Sudoku matrix is chosen at random (so that every Sudoku matrix has equal probability of being chosen). We know two of the squares in this matrix, as shown. What is the probability that the square marked by ? contains the digit $ 3$? $ \setlength{\unitlength}{6mm} \begin{picture}(9,9)(0,0) \multiput(0,0)(1,0){10}{\line(0,1){9}} \multiput(0,0)(0,1){10}{\line(1,0){9}} \linethickness{1.2pt} \multiput(0,0)(3,0){4}{\line(0,1){9}} \multiput(0,0)(0,3){4}{\line(1,0){9}} \put(0,8){\makebox(1,1){1}} \put(1,7){\makebox(1,1){2}} \put(3,6){\makebox(1,1){?}} \end{picture}$

A regular pentagon is drawn in the plane, along with all its diagonals. All its sides and diagonals are extended infinitely in both directions, dividing the plane into regions, some of which are unbounded. An ant starts in the center of the pentagon, and every second, the ant randomly chooses one of the edges of the region it's in, with an equal probability of choosing each edge, and crosses that edge into another region. If the ant enters an unbounded region, it explodes. After first leaving the central region of the pentagon, let $x$ be the expected number of times the ant re-enters the central region before it explodes. Find the closest integer to $100x$.

Let $G$ be an abelian group with $n$ elements, and let \[\{g_1=e,g_2,\dots,g_k\}\subsetneq G\] be a (not necessarily minimal) set of distinct generators of $G.$ A special die, which randomly selects one of the elements $g_1,g_2,\dots,g_k$ with equal probability, is rolled $m$ times and the selected elements are multiplied to produce an element $g\in G.$ Prove that there exists a real number $b\in(0,1)$ such that \[\lim_{m\to\infty}\frac1{b^{2m}}\sum_{x\in G}\left(\mathrm{Prob}(g=x)-\frac1n\right)^2\] is positive and finite.

Carter and Vivian decide to spend their afternoon listing pairs of real numbers, $(a, b).$ Carter wants to find all $(a, b)$ such that $(a, b)$ lie within a circle of radius $6$ centered at $(6, 6).$ Vivian hates circles and would rather find all $(a, b)$ such that $a,$ $b,$ and $6$ can be the side lengths of a triangle. If Carter randomly chooses an $(a, b)$ that satisfies his conditions, then the probability that the pair also satisfies Vivian's conditions can be written in the form $\tfrac{p}{q} + \tfrac{r}{s\pi},$ where $p,$ $q,$ $r,$ and $s$ are positive integers, $p$ and $q$ are relatively prime, and $r$ and $s$ are relatively prime. Find $p + q + r + s.$

If $a,b$ and $c$ are three (not necessarily different) numbers chosen randomly and with replacement from the set $\{1,2,3,4,5 \}$, the probability that $ab+c$ is even is $\textbf{(A)}\ \dfrac{2}{5} \qquad \textbf{(B)}\ \dfrac{59}{125} \qquad \textbf{(C)}\ \dfrac{1}{2} \qquad \textbf{(D)}\ \dfrac{64}{125} \qquad \textbf{(E)}\ \dfrac{3}{5}$

Let $n$ and $k$ be fixed positive integers , and $a$ be arbitrary nonnegative integer . Choose a random $k$-element subset $X$ of $\{1,2,...,k+a\}$ uniformly (i.e., all k-element subsets are chosen with the same probability) and, independently of $X$, choose random n-elements subset $Y$ of $\{1,2,..,k+a+n\}$ uniformly. Prove that the probability $P\left( \text{min}(Y)>\text{max}(X)\right)$ does not depend on $a$.

Starting at $(0,0),$ an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Let $p$ be the probability that the object reaches $(2,2)$ in six or fewer steps. Given that $p$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

Bernardo randomly picks $ 3$ distinct numbers from the set $ \{1,2,3,4,5,6,7,8,9\}$ and arranges them in descending order to form a $ 3$-digit number. Silvia randomly picks $ 3$ distinct numbers from the set $ \{1,2,3,4,5,6,7,8\}$ and also arranges them in descending order to form a $ 3$-digit number. What is the probability that Bernardo's number is larger than Silvia's number? $ \textbf{(A)}\ \frac {47}{72}\qquad \textbf{(B)}\ \frac {37}{56}\qquad \textbf{(C)}\ \frac {2}{3}\qquad \textbf{(D)}\ \frac {49}{72}\qquad \textbf{(E)}\ \frac {39}{56}$

Jake, Jonathan, and Joe are playing a dice game involving polyhedron dice. The dice are as follows: 4 sides, 6 sides, 12 sides, and 20 sides. An n-sided dice has the numbers 1 through n labeled on the sides. Jake starts by selecting a 4-sided die and a 20-sided die. The amount of points that a player gets is the sum of the numbers on the rolled dice. Jonathan then selects a 12-sided die and an 20-sided die. Finally, Joe selects a 20-sided die and a 6-sided die. [list=a] [*]What is the probability that Joe places last? [*]What is the probability that Joe places second? [*]What is the probability that Joe places first? [*]What is the probability that there is a three-way tie? [/list] [i]Proposed by beanielove2[/i]

Aaron takes a square sheet of paper, with one corner labeled $A$. Point $P$ is chosen at random inside of the square and Aaron folds the paper so that points $A$ and $P$ coincide. He cuts the sheet along the crease and discards the piece containing $A$. Let $p$ be the probability that the remaining piece is a pentagon. Find the integer nearest to $100p$. [i]Proposed by Aaron Lin[/i]

The Ultimate Question is a 10-part problem in which each question after the first depends on the answer to the previous problem. As in the Short Answer section, the answer to each (of the 10) problems is a nonnegative integer. You should submit an answer for each of the 10 problems you solve (unlike in previous years). In order to receive credit for the correct answer to a problem, you must also correctly answer $\textit{every one}$ $\textit{of the previous parts}$ $\textit{of the Ultimate Question}$.

Suppose that one of every $500$ people in a certain population has a particular disease, which displays no symptoms. A blood test is available for screening for this disease. For a person who has this disease, the test always turns out positive. For a person who does not have the disease, however, there is a $2\%$ false positive rate; in other words, for such people, $98\%$ of the time the test will turn out negative, but $2\%$ of the time the test will turn out positive and will incorrectly indicate that the person has the disease. Let $p$ be the probability that a person who is chosen at random from the population and gets a positive test result actually has the disease. Which of the following is closest to $p$? $ \textbf{(A)}\ \frac{1}{98}\qquad\textbf{(B)}\ \frac{1}{9}\qquad\textbf{(C)}\ \frac{1}{11}\qquad\textbf{(D)}\ \frac{49}{99}\qquad\textbf{(E)}\ \frac{98}{99}$

Cyrus the frog jumps 2 units in a direction, then 2 more in another direction. What is the probability that he lands less than 1 unit away from his starting position? (I forgot answer choices)

Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require that each player play two games against each of the other's school's players. The match takes place in six rounds, with three games played simultaneously in each round. In how many different ways can the match be scheduled? $\textbf{(A)} \ 540 \qquad \textbf{(B)} \ 600 \qquad \textbf{(C)} \ 720 \qquad \textbf{(D)} \ 810 \qquad \textbf{(E)} \ 900$

Let $k$ be an integer in the interval $[1,99]$. A fair coin is to be flipped $100$ times. Let $$\varepsilon_j =\begin{cases} 1, \text{if the j-th flip is a head} \\ 2, \text{f the j-th flip is a tail}\end{cases}$$ Let $M_k$ denote the probability that there exists a number $i$ such that $k+\varepsilon_1 +...+\varepsilon_i = 100$. How to choose $k$ so as to maximize the probability $M_k$?

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]

Five distinct numbers are drawn successively and at random from the set $\{1, \cdots , n\}$. Show that the probability of a draw in which the first three numbers as well as all five numbers can be arranged to form an arithmetic progression is greater than $\frac{6}{(n-2)^3}$

Nine tiles are numbered $1, 2, 3, \ldots, 9,$ respectively. Each of three players randomly selects and keeps three of the tile, and sums those three values. The probability that all three players obtain an odd sum is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Let $T=\text{TNFTPP}$. Fermi and Feynman play the game $\textit{Probabicloneme}$ in which Fermi wins with probability $a/b$, where $a$ and $b$ are relatively prime positive integers such that $a/b<1/2$. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play $\textit{Probabicloneme}$ so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is $(T-332)/(2T-601)$. Find the value of $a$.

A collection of $2n$ letters contains $2$ each of $n$ different letters. The collection is partitioned into $n$ pairs, each pair containing $2$ letters, which may be the same or different. Denote the number of distinct partitions by $u_n$. (Partitions differing in the order of the pairs in the partition or in the order of the two letters in the pairs are not considered distinct.) Prove that $u_{n+1}=(n+1)u_n - \frac{n(n-1)}{2} u_{n-2}.$ [i]Similar Problem :[/i] A pack of $2n$ cards contains $n$ pairs of $2$ identical cards. It is shuffled and $2$ cards are dealt to each of $n$ different players. Let $p_n$ be the probability that every one of the $n$ players is dealt two identical cards. Prove that $\frac{1}{p_{n+1}}=\frac{n+1}{p_n} + \frac{n(n-1)}{2p_{n-2}}.$

In the Duma there are 1600 delegates, who have formed 16000 committees of 80 persons each. Prove that one can find two committees having no fewer than four common members. [i]A. Skopenkov[/i]

A container contains five red balls. On each turn, one of the balls is selected at random, painted blue, and returned to the container. The expected number of turns it will take before all five balls are colored blue is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.