[b]5.[/b] Let $\xi _{1},\xi _{2},\dots ,\xi _{n},... $ be independent random variables of uniform distribution in $(0,1)$. Show that the distribution of the random variable $\eta _{n}= \sqrt[]{n}\prod_{k=1}^{n}(1-\frac{\xi _{k}}{k}) (n= 1,2,...)$ tends to a limit distribution for $n \to \infty $. [b](P. 6)[/b]

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 $ p_0,p_1,\ldots$ be a probability distribution on the set of nonnegative integers. Select a number according to this distribution and repeat the selection independently until either a zero or an already selected number is obtained. Write the selected numbers in a row in order of selection without the last one. Below this line, write the numbers again in increasing order. Let $ A_i$ denote the event that the number $ i$ has been selected and that it is in the same place in both lines. Prove that the events $ A_i \;(i\equal{}1,2,\ldots)$ are mutually independent, and $ P(A_i)\equal{}p_i$. [i]T. F. Mori[/i]

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 $pq$.

The probability that event $A$ occurs is $\frac{3}{4}$; the probability that event $B$ occurs is $\frac{2}{3}$. Let $p$ be the probability that both $A$ and $B$ occur. The smallest interval necessarily containing $p$ is the interval $ \textbf{(A)}\ \Big[\frac{1}{12},\frac{1}{2}\Big]\qquad\textbf{(B)}\ \Big[\frac{5}{12},\frac{1}{2}\Big]\qquad\textbf{(C)}\ \Big[\frac{1}{2},\frac{2}{3}\Big]\qquad\textbf{(D)}\ \Big[\frac{5}{12},\frac{2}{3}\Big]\qquad\textbf{(E)}\ \Big[\frac{1}{12},\frac{2}{3}\Big]$

Prinstan Trollner and Dukejukem are competing at the game show WASS. Both players spin a wheel which chooses an integer from $1$ to $50$ uniformly at random, and this number becomes their score. Dukejukem then flips a weighted coin that lands heads with probability $\tfrac{3}{5}$. If he flips heads, he adds $1$ to his score. A player wins the game if their score is higher than the other player's score. A player wins the game if their score is higher than the other player's score. The probability Dukejukem defeats the Trollner to win WASS equals $\tfrac{m}{n}$ where $m$ and $n$ are coprime positive integers. Computer $m+n$.

Kelvin and Quinn are collecting trading cards; there are $6$ distinct cards that could appear in a pack. Each pack contains exactly one card, and each card is equally likely. Kelvin buys packs until he has at least one copy of every card, and then he stops buying packs. If Quinn is missing exactly one card, the probability that Kelvin has at least two copies of the card Quinn is missing is expressible as $\tfrac{m}{n}$ for coprime positive integers $m$ and $n$. Determine $m+n$.

On each face of two dice some positive integer is written. The two dice are thrown and the numbers on the top face are added. Determine whether one can select the integers on the faces so that the possible sums are $2,3,4,5,6,7,8,9,10,11,12,13$, all equally likely?

The probability of U2 dismantling an atomic bomb is $11\%$. The probability of Coldplay finding X & Y is $23\%$. If the probability of both events occurring is $ 6\%,$ find the probability that neither occurs.

A gambler plays the following coin-tossing game. He can bet an arbitrary positive amount of money. Then a fair coin is tossed, and the gambler wins or loses the amount he bet depending on the outcome. Our gambler, who starts playing with $ x$ forints, where $ 0<x<2C$, uses the following strategy: if at a given time his capital is $ y<C$, he risks all of it; and if he has $ y>C$, he only bets $ 2C\minus{}y$. If he has exactly $ 2C$ forints, he stops playing. Let $ f(x)$ be the probability that he reaches $ 2C$ (before going bankrupt). Determine the value of $ f(x)$.

Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7$. Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice? $\textbf{(A) } \frac{7}{36} \qquad\textbf{(B) } \frac{5}{24} \qquad\textbf{(C) } \frac{2}{9} \qquad\textbf{(D) } \frac{17}{72} \qquad\textbf{(E) } \frac{1}{4}$

A box contains 11 balls, numbered 1,2,3,....,11. If 6 balls are drawn simultaneously at random, what is the probability that the sum of the numbers on the balls drawn is odd? A. $\frac{100}{231}$ B. $\frac{115}{231}$ C. $\frac{1}{2}$ D. $\frac{118}{231}$ E. $\frac{6}{11}$

In a certain game, Auntie Hall has four boxes $B_1$, $B_2$, $B_3$, $B_4$, exactly one of which contains a valuable gemstone; the other three contain cups of yogurt. You are told the probability the gemstone lies in box $B_n$ is $\frac{n}{10}$ for $n=1,2,3,4$. Initially you may select any of the four boxes; Auntie Hall then opens one of the other three boxes at random (which may contain the gemstone) and reveals its contents. Afterwards, you may change your selection to any of the four boxes, and you win if and only if your final selection contains the gemstone. Let the probability of winning assuming optimal play be $\tfrac mn$, where $m$ and $n$ are relatively prime integers. Compute $100m+n$. [i]Proposed by Evan Chen[/i]

The polynomial $-400x^5+2660x^4-3602x^3+1510x^2+18x-90$ has five rational roots. Suppose you guess a rational number which could possibly be a root (according to the rational root theorem). What is the probability that it actually is a root?

Assume that the number of offspring for every man can be $0,1,\ldots, n$ with with probabilities $p_0,p_1,\ldots,p_n$ independently from each other, where $p_0+p_1+\cdots+p_n=1$ and $p_n\neq 0$. (This is the so-called Galton-Watson process.) Which positive integer $n$ and probabilities $p_0,p_1,\ldots,p_n$ will maximize the probability that the offspring of a given man go extinct in exactly the tenth generation?

A permutation $(a_1, a_2, a_3, \dots, a_{100})$ of $(1, 2, 3, \dots, 100)$ is chosen at random. Denote by $p$ the probability that $a_{2i} > a_{2i - 1}$ for all $i \in \{1, 2, 3, \dots, 50\}$. Compute the number of ordered pairs of positive integers $(a, b)$ satisfying $\textstyle\frac{1}{a^b} = p$. [i]Proposed by Aaron Lin[/i]

A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What is the probability that a reader guessing at random will match all three correctly? $\textbf{(A) }\frac{1}{9}\qquad\textbf{(B) }\frac{1}{6}\qquad\textbf{(C) }\frac{1}{4}\qquad\textbf{(D) }\frac{1}{3}\qquad \textbf{(E) }\frac{1}{2}$

Professor Gamble buys a lottery ticket, which requires that he pick six different integers from $ 1$ through $ 46$, inclusive. He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer. It so happens that the integers on the winning ticket have the same property--- the sum of the base-ten logarithms is an integer. What is the probability that Professor Gamble holds the winning ticket? $ \textbf{(A)}\ 1/5 \qquad \textbf{(B)}\ 1/4 \qquad \textbf{(C)}\ 1/3 \qquad \textbf{(D)}\ 1/2 \qquad \textbf{(E)}\ 1$

Written on a blackboard is the polynomial $x^2+x+2014$. Calvin and Hobbes take turns alternately (starting with Calvin) in the following game. At his turn, Calvin should either increase or decrease the coefficient of $x$ by $1$. And at this turn, Hobbes should either increase or decrease the constant coefficient by $1$. Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots. Prove that Calvin has a winning stratergy.

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

Ana and Banana play a game. First, Ana picks a real number $p$ with $0 \le p \le 1$. Then, Banana picks an integer $h$ greater than $1$ and creates a spaceship with $h$ hit points. Now every minute, Ana decreases the spaceship's hit points by $2$ with probability $1-p$, and by $3$ with probability $p$. Ana wins if and only if the number of hit points is reduced to exactly $0$ at some point (in particular, if the spaceship has a negative number of hit points at any time then Ana loses). Given that Ana and Banana select $p$ and $h$ optimally, compute the integer closest to $1000p$. [i]Proposed by Lewis Chen[/i]

There is a test for the dangerous bifurcation virus that is $ 99\%$ accurate. In other words, if someone has the virus, there is a $ 99\%$ chance that the test will be positive, and if someone does not have it, then there is a $ 99\%$ chance the test will be negative. Assume that exactly $ 1\%$ of the general population has the virus. Given an individual that has tested positive from this test, what is the probability that he or she actually has the disease? Express your answer as a percentage.

A cube with $3$-inch edges is to be constructed from $27$ smaller cubes with $1$-inch edges. Twenty-one of the cubes are colored red and $6$ are colored white. If the $3$-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white? $\textbf{(A) }\frac{5}{54}\qquad\textbf{(B) }\frac{1}{9}\qquad\textbf{(C) }\frac{5}{27}\qquad\textbf{(D) }\frac{2}{9}\qquad \textbf{(E) }\frac{1}{3}$

We call a subset $S$ of vertices of graph $G$, $2$-dominating, if and only if for every vertex $v\notin S,v\in G$, $v$ has at least two neighbors in $S$. prove that every $r$-regular $(r\ge3)$ graph has a $2$-dominating set with size at most $\frac{n(1+\ln(r))}{r}$.(15 points) time of this exam was 3 hours

A student puts $2010$ red balls and $1957$ blue balls into a box. Weiqing draws randomly from the box one ball at a time without replacement. She wins if, at anytime, the total number of blue balls drawn is more than the total number of red balls drawn. Assuming Weiqing keeps drawing balls until she either wins or runs out, ompute the probability that she eventually wins.