A cube is assembled with $27$ white cubes. The larger cube is then painted black on the outside and disassembled. A blind man reassembles it. What is the probability that the cube is now completely black on the outside? Give an approximation of the size of your answer.

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 $m + n$.

If $ p \geq 5$ is a prime number, then $ 24$ divides $ p^2 \minus{} 1$ without remainder $ \textbf{(A)}\ \text{never} \qquad \textbf{(B)}\ \text{sometimes only} \qquad \textbf{(C)}\ \text{always} \qquad$ $ \textbf{(D)}\ \text{only if } p \equal{}5 \qquad \textbf{(E)}\ \text{none of these}$

Let $p_n(k)$ be the number of permutations of the set $\{1,2,3,\ldots,n\}$ which have exactly $k$ fixed points. Prove that $\sum_{k=0}^nk p_n(k)=n!$.[i](IMO Problem 1)[/i] [b][i]Original formulation [/i][/b] Let $S$ be a set of $n$ elements. We denote the number of all permutations of $S$ that have exactly $k$ fixed points by $p_n(k).$ Prove: (a) $\sum_{k=0}^{n} kp_n(k)=n! \ ;$ (b) $\sum_{k=0}^{n} (k-1)^2 p_n(k) =n! $ [i]Proposed by Germany, FR[/i]

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?

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?

P6. It is given the integer $Y$ with $Y = 2018 + 20118 + 201018 + 2010018 + \cdots + 201 \underbrace{00 \ldots 0}_{\textrm{100 digits}} 18.$ Determine the sum of all the digits of such $Y$. (It is implied that $Y$ is written with a decimal representation.) P7. Three groups of lines divides a plane into $D$ regions. Every pair of lines in the same group are parallel. Let $x, y$ and $z$ respectively be the number of lines in groups 1, 2, and 3. If no lines in group 3 go through the intersection of any two lines (in groups 1 and 2, of course), then the least number of lines required in order to have more than 2018 regions is .... P8. It is known a frustum $ABCD.EFGH$ where $ABCD$ and $EFGH$ are squares with both planes being parallel. The length of the sides of $ABCD$ and $EFGH$ respectively are $6a$ and $3a$, and the height of the frustum is $3t$. Points $M$ and $N$ respectively are intersections of the diagonals of $ABCD$ and $EFGH$ and the line $MN$ is perpendicular to the plane $EFGH$. Construct the pyramids $M.EFGH$ and $N.ABCD$ and calculate the volume of the 3D figure which is the intersection of pyramids $N.ABCD$ and $M.EFGH$. P9. Look at the arrangement of natural numbers in the following table. The position of the numbers is determined by their row and column numbers, and its diagonal (which, the sequence of numbers is read from the bottom left to the top right). As an example, the number $19$ is on the 3rd row, 4th column, and on the 6th diagonal. Meanwhile the position of the number $26$ is on the 3rd row, 5th column, and 7th diagonal. (Image should be placed here, look at attachment.) a) Determine the position of the number $2018$ based on its row, column, and diagonal. b) Determine the average of the sequence of numbers whose position is on the "main diagonal" (quotation marks not there in the first place), which is the sequence of numbers read from the top left to the bottom right: 1, 5, 13, 25, ..., which the last term is the largest number that is less than or equal to $2018$. P10. It is known that $A$ is the set of 3-digit integers not containing the digit $0$. Define a [i]gadang[/i] number to be the element of $A$ whose digits are all distinct and the digits contained in such number are not prime, and (a [i]gadang[/i] number leaves a remainder of 5 when divided by 7. If we pick an element of $A$ at random, what is the probability that the number we picked is a [i]gadang[/i] number?

Arnold is studying the prevalence of three health risk factors, denoted by A, B, and C. within a population of men. For each of the three factors, the probability that a randomly selected man in the population as only this risk factor (and none of the others) is 0.1. For any two of the three factors, the probability that a randomly selected man has exactly two of these two risk factors (but not the third) is 0.14. The probability that a randomly selected man has all three risk factors, given that he has A and B is $\tfrac{1}{3}$. The probability that a man has none of the three risk factors given that he does not have risk factor A is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

When rolling a certain unfair six-sided die with faces numbered $1, 2, 3, 4, 5$, and $6$, the probability of obtaining face $F$ is greater than $\frac{1}{6}$, the probability of obtaining the face opposite is less than $\frac{1}{6}$, the probability of obtaining any one of the other four faces is $\frac{1}{6}$, and the sum of the numbers on opposite faces is $7$. When two such dice are rolled, the probability of obtaining a sum of $7$ is $\frac{47}{288}$. Given that the probability of obtaining face $F$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Ali, Bonnie, Carlo, and Dianna are going to drive together to a nearby theme park. The car they are using has $4$ seats: $1$ Driver seat, $1$ front passenger seat, and $2$ back passenger seat. Bonnie and Carlo are the only ones who know how to drive the car. How many possible seating arrangements are there? $\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 24$

You roll three fair six-sided dice. Given that the highest number you rolled is a $5$, the expected value of the sum of the three dice can be written as $\tfrac ab$ in simplest form. Find $a+b$.

Let $b$ be the probability that the cards are from different suits. Compute $\lfloor1000b\rfloor$.

Thee tiles are marked $ X$ and two other tiles are marked $ O$. The five tiles are randomly arranged in a row. What is the probability that the arrangement reads $ XOXOX$? $ \textbf{(A)}\ \frac{1}{12}\qquad \textbf{(B)}\ \frac{1}{10}\qquad \textbf{(C)}\ \frac{1}{6}\qquad \textbf{(D)}\ \frac{1}{4}\qquad \textbf{(E)}\ \frac{1}{3}$

A real number $ to $ is randomly and uniformly chosen from the $ [- 3,4] $ interval. What is the probability that all roots of the polynomial $ x ^ 3 + ax ^ 2 + ax + 1 $ are real?

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 $a+b$.

Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. What is the probability that Angie and Carlos are seated opposite each other? $ \textbf{(A)} \frac14 \qquad\textbf{(B)} \frac13 \qquad\textbf{(C)} \frac12 \qquad\textbf{(D)} \frac23 \qquad\textbf{(E)} \frac34 $

An equilateral triangle is tiled with $n^2$ smaller congruent equilateral triangles such that there are $n$ smaller triangles along each of the sides of the original triangle. For each of the small equilateral triangles, we randomly choose a vertex $V$ of the triangle and draw an arc with that vertex as center connecting the midpoints of the two sides of the small triangle with $V$ as an endpoint. Find, with proof, the expected value of the number of full circles formed, in terms of $n.$ [img]http://s3.amazonaws.com/classroom.artofproblemsolving.com/Images/Transcripts/497b4e1ef5043a84b433a5c52c4be3ae.png[/img]

Every time Josh and Ron tap their screens, one of three emojis appears, each with equal probability: barbecue, bacon, or burger. Josh taps his screen until he gets a sequence of barbecue, bacon, and burger consecutively (in that specific order.) Ron taps his screen until he gets a sequence of three bacons in a row. Let $M$ and $N$ be the expected number of times Josh and Ron tap their screens, respectively. What is $|M-N|$?

A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once? $ \textbf{(A) } \frac {1}{2187} \qquad \textbf{(B) } \frac {1}{729} \qquad \textbf{(C) } \frac {2}{243} \qquad \textbf{(D) } \frac {1}{81} \qquad \textbf{(E) } \frac {5}{243}$

suppose that $S_{\mathbb N}$ is the set of all permutations of natural numbers. finite permutations are a subset of $S_{\mathbb N}$ that behave like the identity permutation from somewhere. in other words bijective functions like $\pi: \mathbb N \longrightarrow \mathbb N$ that only for finite natural numbers $i$, $\pi(i)\neq i$. prove that we cannot put probability measure that is countably additive on $\wp(S_{\mathbb N})$ (family of all the subsets of $S_{\mathbb N}$) that is invarient under finite permutations.

Consider an unbiased coin which is tossed infinitely many times. Let $A_n$ be the event that no two successive heads occur in the first $n$ tosses of this experiment. Then which of the following is incorrect : (A) $\lim_{n \to \infty} P(A_n)=0$ (B) $\lim_{n \to \infty}3^n P(A_n)=0$ (C) $2^nP(A_n) +2^{n+1}P(A_{n+1})=2^{n+2}P(A_{n+2}$ (D) $\lim_{n \to \infty} \frac{P(A_n)}{P(A_{n+1})}$ is lesser than $1.2$

In a test, $3n$ students participate, who are located in three rows of $n$ students in each. The students leave the test room one by one. If $N_1(t), N_2(t), N_3(t)$ denote the numbers of students in the first, second, and third row respectively at time $t$, find the probability that for each t during the test, \[|N_i(t) - N_j(t)| < 2, i \neq j, i, j = 1, 2, \dots .\]

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 $m.$

The New York Times Mini Crossword is a $5\times5$ grid with the top left and bottom right corners shaded. Each row and column has a clue given (so that there are $10$ clues total). Jeffrey has a $\frac{1}{2}$ chance of knowing the answer to each clue. What is the probability that he can fill in every unshaded square in the crossword? [asy] size(4cm); for (int i = 0; i < 6; ++i) {draw((i,0)--(i,5)); draw((0,i)--(5,i));} fill((0,4)--(1,4)--(1,5)--(0,5)--cycle, black); fill((5,0)--(5,1)--(4,1)--(4,0)--cycle, black); [/asy] [i]2018 CCA Math Bonanza Individual Round #8[/i]

A coin is biased in such a way that on each toss the probability of heads is $\frac{2}{3}$ and the probability of tails is $\frac{1}{3}$. The outcomes of the tosses are independent. A player has the choice of playing Game A or Game B. In Game A she tosses the coin three times and wins if all three outcomes are the same. In Game B she tosses the coin four times and wins if both the outcomes of the first and second tosses are the same and the outcomes of the third and fourth tosses are the same. How do the chances of winning Game A compare to the chances of winning Game B? $\textbf{(A)} \text{ The probability of winning Game A is }\frac{4}{81}\text{ less than the probability of winning Game B.} $ $\textbf{(B)} \text{ The probability of winning Game A is }\frac{2}{81}\text{ less than the probability of winning Game B.}$ $\textbf{(C)} \text{ The probabilities are the same.}$ $\textbf{(D)} \text{ The probability of winning Game A is }\frac{2}{81}\text{ greater than the probability of winning Game B.}$ $\textbf{(E)} \text{ The probability of winning Game A is }\frac{4}{81}\text{ greater than the probability of winning Game B.}$