Let $S = \{(x, y) : x, y \in \{1, 2, 3, \dots, 2012\}\}$. For all points $(a, b)$, let $N(a, b) = \{(a - 1, b), (a + 1, b), (a, b - 1), (a, b + 1)\}$. Kathy constructs a set $T$ by adding $n$ distinct points from $S$ to $T$ at random. If the expected value of $\displaystyle \sum_{(a, b) \in T} | N(a, b) \cap T |$ is 4, then compute $n$. [i]Proposed by Lewis Chen[/i]

Palmer and James work at a dice factory, placing dots on dice. Palmer builds his dice correctly, placing the dots so that $1$, $2$, $3$, $4$, $5$, and $6$ dots are on separate faces. In a fit of mischief, James places his $21$ dots on a die in a peculiar order, putting some nonnegative integer number of dots on each face, but not necessarily in the correct configuration. Regardless of the configuration of dots, both dice are unweighted and have equal probability of showing each face after being rolled. Then Palmer and James play a game. Palmer rolls one of his normal dice and James rolls his peculiar die. If they tie, they roll again. Otherwise the person with the larger roll is the winner. What is the maximum probability that James wins? Give one example of a peculiar die that attains this maximum probability.

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

From the set of all permutations $f$ of $\{1, 2, ... , n\}$ that satisfy the condition: $f(i) \geq i-1$ $i=1,...,n$ one is chosen uniformly at random. Let $p_n$ be the probability that the chosen permutation $f$ satisfies $f(i) \leq i+1$ $i=1,...,n$ Find all natural numbers $n$ such that $p_n > \frac{1}{3}$.

In a drawer, there are at most $ 2009$ balls, some of them are white, the rest are blue, which are randomly distributed. If two balls were taken at the same time, then the probability that the balls are both blue or both white is $ \frac12$. Determine the maximum amount of white balls in the drawer, such that the probability statement is true?

A robot on the number line starts at $1$. During the first minute, the robot writes down the number $1$. Each minute thereafter, it moves by one, either left or right, with equal probability. It then multiplies the last number it wrote by $n/t$, where $n$ is the number it just moved to, and $t$ is the number of minutes elapsed. It then writes this number down. For example, if the robot moves right during the second minute, it would write down $2/2=1$. Find the expected sum of all numbers it writes down, given that it is finite. [i]Proposed by Ethan Tan[/i]

Let $S$ be a randomly selected four-element subset of $\{1, 2, 3, 4, 5, 6, 7, 8\}$. Let $m$ and $n$ be relatively prime positive integers so that the expected value of the maximum element in $S$ is $\dfrac{m}{n}$. Find $m + n$.

Mark's teacher is randomly pairing his class of $16$ students into groups of $2$ for a project. What is the probability that Mark is paired up with his best friend, Mike? (There is only one Mike in the class) [i]2015 CCA Math Bonanza Individual Round #3[/i]

The two spinners shown are spun once and each lands on one of the numbered sectors. What is the probability that the sum of the numbers in the two sectors is prime? [asy]unitsize(30); draw(unitcircle); draw((0,0)--(0,-1)); draw((0,0)--(cos(pi/6),sin(pi/6))); draw((0,0)--(-cos(pi/6),sin(pi/6))); label("$1$",(0,.5)); label("$3$",((cos(pi/6))/2,(-sin(pi/6))/2)); label("$5$",(-(cos(pi/6))/2,(-sin(pi/6))/2));[/asy] [asy]unitsize(30); draw(unitcircle); draw((0,0)--(0,-1)); draw((0,0)--(cos(pi/6),sin(pi/6))); draw((0,0)--(-cos(pi/6),sin(pi/6))); label("$2$",(0,.5)); label("$4$",((cos(pi/6))/2,(-sin(pi/6))/2)); label("$6$",(-(cos(pi/6))/2,(-sin(pi/6))/2));[/asy] $ \textbf{(A)}\ \frac {1}{2} \qquad \textbf{(B)}\ \frac {2}{3} \qquad \textbf{(C)}\ \frac {3}{4} \qquad \textbf{(D)}\ \frac {7}{9} \qquad \textbf{(E)}\ \frac {5}{6}$

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}$

a government has decided to help it's people by giving them $n$ coupons for $n$ fundamental things, but because of being unmanaged, the giving of the coupons to the people is random. in each time that a person goes to the office to get a coupon, the office manager gives him one of the $n$ coupons randomly and with the same probability. It's obvious that in this system a person may get a coupon that he had it before. suppose that $X_n$ is the random varieble of the first time that a person gets all of the $n$ coupons. show that $\frac{X_n}{n ln(n)}$ in probability converges to $1$.

Let $(\Omega, \mathcal A, P)$ be a probability space, and let $(X_n, \mathcal F_n)$ be an adapted sequence in $(\Omega, \mathcal A, P)$ (that is, for the $\sigma$-algebras $\mathcal F_n$, we have $\mathcal F_1\subseteq \mathcal F_2\subseteq \dots \subseteq \mathcal A$, and for all $n$, $X_n$ is an $\mathcal F_n$-measurable and integrable random variable). Assume that $$\mathrm E (X_{n+1} \mid \mathcal F_n )=\frac12 X_n+\frac12 X_{n-1}\,\,\,\,\, (n=2, 3, \ldots )$$ Prove that $\mathrm{sup}_n \mathrm{E}|X_n|<\infty$ implies that $X_n$ converges with probability one as $n\to\infty$. [I. Fazekas]

Let $B$ and $D$ be two points chosen independently and uniformly at random from the unit sphere in 3D space centered at a point $A$ (this unit sphere is the set of all points in $\mathbb{R}^3$ a distance of $1$ away from $A$). Compute the expected value of $\sin^2\angle DAB$.

Team Stanford has a $ \frac{1}{3}$ chance of winning any given math contest. If Stanford competes in 4 contests this quarter, what is the probability that the team will win at least once?

A point $ P$ is randomly selected from the rectangular region with vertices $ (0, 0)$, $ (2, 0)$, $ (2, 1)$, $ (0, 1)$. What is the probability that $ P$ is closer to the origin than it is to the point $ (3, 1)$? $ \textbf{(A)}\ \frac{1}{2} \qquad \textbf{(B)}\ \frac{2}{3} \qquad \textbf{(C)}\ \frac{3}{4} \qquad \textbf{(D)}\ \frac{4}{5} \qquad \textbf{(E)}\ 1$

Kathryn has a crush on Joe. Dressed as Catwoman, she attends the same school Halloween party as Joe, hoping he will be there. If Joe gets beat up, Kathryn will be able to help Joe, and will be able to tell him how much she likes him. Otherwise, Kathryn will need to get her hipster friend, Max, who is DJing the event, to play Joe’s favorite song, “Pieces of Me” by Ashlee Simpson, to get him out on the dance floor, where she’ll also be able to tell him how much she likes him. Since playing the song would be in flagrant violation of Max’s musical integrity as a DJ, Kathryn will have to bribe him to play the song. For every $\$10$ she gives Max, the probability of him playing the song goes up $10\%$ (from $0\%$ to $10\%$ for the first $\$10$, from $10\%$ to $20\%$ for the next $\$10$, all the way up to $100\%$ if she gives him $\$100$). Max only accepts money in increments of $\$10$. How much money should Kathryn give to Max to give herself at least a $65\%$ chance of securing enough time to tell Joe how much she likes him?

Four different positive integers less than 10 are chosen randomly. What is the probability that their sum is odd?

The world-renowned Marxist theorist [i]Joric[/i] is obsessed with both mathematics and social egalitarianism. Therefore, for any decimal representation of a positive integer $n$, he tries to partition its digits into two groups, such that the difference between the sums of the digits in each group be as small as possible. Joric calls this difference the [i]defect[/i] of the number $n$. Determine the average value of the defect (over all positive integers), that is, if we denote by $\delta(n)$ the defect of $n$, compute \[\lim_{n \rightarrow \infty}\frac{\sum_{k = 1}^{n}\delta(k)}{n}.\] [i]Iurie Boreico[/i]

Tina randomly selects two distinct numbers from the set $ \{1,2,3,4,5\}$ and Sergio randomly selects a number from the set $ \{1,2,...,10\}$. The probability that Sergio's number is larger than the sum of the two numbers chosen by Tina is $ \textbf{(A)}\ 2/5 \qquad \textbf{(B)}\ 9/20 \qquad \textbf{(C)}\ 1/2\qquad \textbf{(D)}\ 11/20 \qquad \textbf{(E)}\ 24/25$

A jar contains 4 blue marbles, 3 green marbles, and 5 red marbles. If Helen reaches in the jar and selects a marble at random, then the probability that she selects a red marble can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Cast a dice $n$ times. Denote by $X_1,\ X_2,\ \cdots ,\ X_n$ the numbers shown on each dice. Define $Y_1,\ Y_2,\ \cdots,\ Y_n$ by \[Y_1=X_1,\ Y_k=X_k+\frac{1}{Y_{k-1}}\ (k=2,\ \cdots,\ n)\] Find the probability $p_n$ such that $\frac{1+\sqrt{3}}{2}\leq Y_n\leq 1+\sqrt{3}.$ 35 points

A small apartment building has four doors, with door numbers $1, 2, 3, 4.$ John has $2^4-1=15$ keys, label with of possible nonempty subsets of $\{1,2,3,4\}$, but he forgot which key is which. If an element on the key matches the door number, the key can open the door (e.g. key $\{1,2,4\}$ can open Door 4). He picks a key at random and tries to open Door 1, which fails, so he discards it. John then randomly picks one of his remaining 14 keys and tries to open Door 2, but it doesn't open, so he throws away that key as well. He then randomly selects one of the remaining 13 keys, and tests it on Door 3. What is the probability that it will open? [i]Proposed by dantx5[/i]

Four boys and four girls each bring one gift to a Christmas gift exchange. On a sheet of paper, each boy randomly writes down the name of one girl, and each girl randomly writes down the name of one boy. At the same time, each person passes their gift to the person whose name is written on their sheet. Determine the probability that [i]both[/i] of these events occur: [list] [*] (i) Each person receives exactly one gift; [*] (ii) No two people exchanged presents with each other (i.e., if $A$ gave his gift to $B$, then $B$ did not give her gift to $A$).[/list]

The Cincinnati Reals are playing the Houston Alphas in the last game of the Swirled Series. The Alphas are leading by $1$ run in the bottom of the $9\text{th}$ (last) inning, and the Reals are at bat. Each batter has a $\dfrac{1}{3}$ chance of hitting a single and a $\dfrac{2}{3}$ chance of making an out. If the Reals hit $5$ or more singles before they make $3$ outs, they will win. If the Reals hit exactly $4$ singles before they make $3$ outs, they will tie the game and send it into extra innings, and they will have a $\dfrac{3}{5}$ chance of eventually winning the game (since they have the added momentum of coming from behind). If the Reals hit fewer than $4$ singles, they will LOSE! What is the probability that the Alphas hold off the Reals and win, sending the packed Alphadome into a frenzy? Express the answer as a fraction.

$p$ passengers get on a train with $n$ wagons. Find the probability of being at least one passenger at each wagon.