suppose that $0\le p \le 1$ and we have a wooden square with side length $1$. in the first step we cut this square into $4$ smaller squares with side length $\frac{1}{2}$ and leave each square with probability $p$ or take it with probability $1-p$. in the next step we cut every remaining square from the previous step to $4$ smaller squares (as above) and take them with probability $1-p$. it's obvios that at the end what remains is a subset of the first square. [b]a)[/b] show that there exists a number $0<p_0<1$ such that for $p>p_0$ the probability that the remainig set is not empty is positive and for $p<p_0$ this probability is zero. [b]b)[/b] show that for every $p\neq 1$ with probability $1$, the remainig set has size zero. [b]c)[/b] for this statement that the right side of the square is connected to the left side of the square with a path, write anything that you can.

Select numbers $ a$ and $ b$ between $ 0$ and $ 1$ independently and at random, and let $ c$ be their sum. Let $ A, B$ and $ C$ be the results when $ a, b$ and $ c$, respectively, are rounded to the nearest integer. What is the probability that $ A \plus{} B \equal{} C$? $ \textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac13 \qquad \textbf{(C)}\ \frac12 \qquad \textbf{(D)}\ \frac23 \qquad \textbf{(E)}\ \frac34$

10 students are arranged in a row. Every minute, a new student is inserted in the row (which can occur in the front and in the back as well, hence $11$ possible places) with a uniform $\tfrac{1}{11}$ probability of each location. Then, either the frontmost or the backmost student is removed from the row (each with a $\tfrac{1}{2}$ probability). Suppose you are the eighth in the line from the front. The probability that you exit the row from the front rather than the back is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $100m+n$. [i]Proposed by Lewis Chen[/i]

Two concentric circles have radii $1$ and $2$. Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle? $\textbf{(A) }\frac{1}{6}\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }\frac{2-\sqrt{2}}{2}\qquad\textbf{(D) }\frac{1}{3}\qquad\textbf{(E) }\frac{1}{2}\qquad$

In a game, there are three indistinguishable boxes; one box contains two red balls, one contains two blue balls, and the last contains one ball of each color. To play, Raj first predicts whether he will draw two balls of the same color or two of different colors. Then, he picks a box, draws a ball at random, looks at the color, and replaces the ball in the same box. Finally, he repeats this; however, the boxes are not shuffled between draws, so he can determine whether he wants to draw again from the same box. Raj wins if he predicts correctly; if he plays optimally, what is the probability that he will win?

Suppose that $728$ coins are set on a table, all facing heads up at first. For each iteration, we randomly choose $314$ coins and flip them (from heads to tails or vice versa). Let $a/b$ be the expected number of heads after we finish $4001$ iterations, where $a$ and $b$ are relatively prime. Find $a + b$ mod $10000$.

What is the probability of having $2$ adjacent white balls or $2$ adjacent blue balls in a random arrangement of $3$ red, $2$ white and $2$ blue balls? $ \textbf{(A)}\ \dfrac{2}{5} \qquad\textbf{(B)}\ \dfrac{3}{7} \qquad\textbf{(C)}\ \dfrac{16}{35} \qquad\textbf{(D)}\ \dfrac{10}{21} \qquad\textbf{(E)}\ \dfrac{5}{14} $

A hotel packed breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls and once wrapped, the rolls were indistinguishable from one another. She then randomly put three rolls in a bag for each of the guests. Given that the probability each guest got one roll of each type is $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m+n$.

The Miami Heat and the San Antonio Spurs are playing a best-of-five series basketball championship, in which the team that first wins three games wins the whole series. Assume that the probability that the Heat wins a given game is $x$ (there are no ties). The expected value for the total number of games played can be written as $f(x)$, with $f$ a polynomial. Find $f(-1)$.

As part of his effort to take over the world, Edward starts producing his own currency. As part of an effort to stop Edward, Alex works in the mint and produces $1$ counterfeit coin for every $99$ real ones. Alex isn't very good at this, so none of the counterfeit coins are the right weight. Since the mint is not perfect, each coin is weighed before leaving. If the coin is not the right weight, then it is sent to a lab for testing. The scale is accurate $95\%$ of the time, $5\%$ of all the coins minted are sent to the lab, and the lab's test is accurate $90\%$ of the time. If the lab says a coin is counterfeit, what is the probability that it really is?

For each permutation $ a_1, a_2, a_3, \ldots,a_{10}$ of the integers $ 1,2,3,\ldots,10,$ form the sum \[ |a_1 \minus{} a_2| \plus{} |a_3 \minus{} a_4| \plus{} |a_5 \minus{} a_6| \plus{} |a_7 \minus{} a_8| \plus{} |a_9 \minus{} a_{10}|.\] The average value of all such sums can be written in the form $ p/q,$ where $ p$ and $ q$ are relatively prime positive integers. Find $ p \plus{} q.$

A frog is jumping on $N$ stones which are numbered from $1$ to $N$ from left to right. The frog is jumping to the previous stone (to the left) with probability $p$ and is jumping to the next stone (to the right) with probability $1-p$. If the frog has jumped to the left from the leftmost stone or to the right from the rightmost stone, it will fall into the water. The frog is initially on the leftmost stone. If $p< \tfrac 13$, show that the frog will fall into the water from the rightmost stone with a probability higher than $\tfrac 12$.

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 $s(n)$.

An urn was filled with three balls by the following procedure: it was thrown a coin three times, inserting, each time a white ball came up heads, and every time tails came up, a black ball. We draw from this urn, four times consecutive, one ball; we return it to the urn before the next extraction. Which is the probability that in the four extractions a cue ball is obtained?

PUMaCDonalds, a newly-opened fast food restaurant, has 5 menu items. If the first 4 customers each choose one menu item at random, the probability that the 4th customer orders a previously unordered item is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

A circle of radius 1 is randomly placed in a 15-by-36 rectangle $ABCD$ so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal $AC$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Ten women sit in $ 10$ seats in a line. All of the $ 10$ get up and then reseat themselves using all $ 10$ seats, each sitting in the seat she was in before or a seat next to the one she occupied before. In how many ways can the women be reseated? $ \textbf{(A)}\ 89\qquad \textbf{(B)}\ 90\qquad \textbf{(C)}\ 120\qquad \textbf{(D)}\ 2^{10}\qquad \textbf{(E)}\ 2^2 3^8$

Let $S$ be the set of integers between $1$ and $2^{40}$ whose binary expansions have exactly two $1$'s. If a number is chosen at random from $S$, the probability that it is divisible by $9$ is $p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

In the game of "Fingers", $N$ players stand in a circle and simultaneously thrust out their right hands, each with a certain number of fingers showing. The total number of fingers shown is counted out round the circle from the leader, and the player on whom the count stops is the winner. How large must $N$ be for a suitably chosen group of $N/10$ players to contain a winner with probability at least $0.9$? How does the probability that the leader wins behave as $N\to\infty$?

Prove that there exists a positive $ c$ such that for every positive integer $ N$ among any $ N$ positive integers not exceeding $ 2N$ there are two numbers whose greatest common divisor is greater than $ cN$.

Pyramid $EARLY$ is placed in $(x,y,z)$ coordinates so that $E=(10,10,0),A=(10,-10,0)$, $R=(-10,-10,0)$, $L=(-10,10,0)$, and $Y=(0,0,10)$. Tunnels are drilled through the pyramid in such a way that one can move from $(x,y,z)$ to any of the $9$ points $(x,y,z-1)$, $(x\pm 1,y,z-1)$, $(x,y\pm 1, z-1)$, $(x\pm 1, y\pm 1, z-1)$. Sean starts at $Y$ and moves randomly down to the base of the pyramid, choosing each of the possible paths with probability $\dfrac{1}{9}$. What is the probability that he ends up at the point $(8,9,0)$?

Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability it is tossed into bin $i$ is $2^{-i}$ for $i = 1, 2, 3, \ldots$. More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins $3$, $17$, and $10$.) What is $p+q$? $\textbf{(A)}\ 55 \qquad\textbf{(B)}\ 56 \qquad\textbf{(C)}\ 57 \qquad\textbf{(D)}\ 58 \qquad\textbf{(E)}\ 59$

Let $x$ be a real number selected uniformly at random between 100 and 200. If $\lfloor {\sqrt{x}} \rfloor = 12$, find the probability that $\lfloor {\sqrt{100x}} \rfloor = 120$. ($\lfloor {v} \rfloor$ means the greatest integer less than or equal to $v$.) $\text{(A)} \ \frac{2}{25} \qquad \text{(B)} \ \frac{241}{2500} \qquad \text{(C)} \ \frac{1}{10} \qquad \text{(D)} \ \frac{96}{625} \qquad \text{(E)} \ 1$

Given integers \( m, n \geq 2 \), the points \( A_1, A_2, \dots, A_n \) are chosen independently and uniformly at random on a circle of circumference \( 1 \). That is, for each \( i = 1, \dots, n \), for any \( x \in (0,1) \), and for any arc \( \mathcal{C} \) of length \( x \) on the circle, we have $\mathbb{P}(A_i \in \mathcal{C}) = x$. What is the probability that there exists an arc of length \( \frac{1}{m} \) on the circle that contains all the points \( A_1, A_2, \dots, A_n \)?

Three red beads, two white beads, and one blue bead are placed in a line in random order. What is the probability that no two neighboring beads are the same color? $ \textbf{(A)}\ \frac{1}{12} \qquad \textbf{(B)}\ \frac{1}{10} \qquad \textbf{(C)}\ \frac{1}{6} \qquad \textbf{(D)}\ \frac{1}{3} \qquad \textbf{(E)}\ \frac{1}{2}$