Three distinct vertices are chosen at random from the vertices of a given regular polygon of $ (2n\plus{}1)$ sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen random points?

Two cards are dealt from a deck of four red cards labeled $A$, $B$, $C$, $D$ and four green cards labeled $A$, $B$, $C$, $D$. A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair? $\textbf{(A)}\ \frac{2}{7} \qquad \textbf{(B)}\ \frac{3}{8} \qquad \textbf{(C)}\ \frac{1}{2} \qquad \textbf{(D)}\ \frac{4}{7} \qquad \textbf{(E)}\ \frac{5}{8}$

A Vandal and a Moderator are editing a Wikipedia article. The article originally is error-free. Each day, the Vandal introduces one new error into the Wikipedia article. At the end of the day, the moderator checks the article and has a $ 2/3$ chance of catching each individual error still in the article. After $ 3$ days, what is the probability that the article is error-free?

Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices of three being equally likely - and are sent off to slay a troublesome dragon. Let $P$ be the probability that at least two of the three had been sitting next to each other. If $P$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?

Two circles of radius 1 are to be constructed as follows. The center of circle $ A$ is chosen uniformly and at random from the line segment joining $ (0,0)$ and $ (2,0)$. The center of circle $ B$ is chosen uniformly and at random, and independently of the first choice, from the line segment joining $ (0,1)$ to $ (2,1)$. What is the probability that circles $ A$ and $ B$ intersect? $ \textbf{(A)} \; \frac{2\plus{}\sqrt{2}}{4} \qquad \textbf{(B)} \; \frac{3\sqrt{3}\plus{}2}{8} \qquad \textbf{(C)} \; \frac{2 \sqrt{2} \minus{} 1}{2} \qquad \textbf{(D)} \; \frac{2\plus{}\sqrt{3}}{4} \qquad \textbf{(E)} \; \frac{4 \sqrt{3} \minus{} 3}{4}$

Let $X_1, X_2$ be independent, identically distributed random variables such that $X_i\geq 0$ for all $i$. Let $\mathrm EX_i=m$, $\mathrm{Var} (X_i)=\sigma ^2<\infty$. Show that, for all $0<\alpha\leq 1$ $$\lim_{n\to\infty} n\,\mathrm{Var} \left( \left[ \frac{X_1+\ldots +X_n}{n}\right] ^\alpha\right)=\frac{\alpha ^ 2 \sigma ^ 2}{m^{2(1-\alpha)}}$$ [Gy. Michaletzki]

Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area? $ \textbf{(A)} \ \frac{553}{715} \qquad \textbf{(B)} \ \frac{443}{572} \qquad \textbf{(C)} \ \frac{111}{143} \qquad \textbf{(D)} \ \frac{81}{104} \qquad \textbf{(E)} \ \frac{223}{286}$

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.

$20$ players are playing in a Super Mario Smash Bros. Melee tournament. They are ranked $1-20$, and player $n$ will always beat player $m$ if $n<m$. Out of all possible tournaments where each player plays $18$ distinct other players exactly once, one is chosen uniformly at random. Find the expected number of pairs of players that win the same number of games.

Done working on his sand castle design, Joshua sits down and starts rolling a $12$-sided die he found when cleaning the storage shed. He rolls and rolls and rolls, and after $17$ rolls he finally rolls a $1$. Just $3$ rolls later he rolls the first $2\textit{ after}$ that first roll of $1$. $11$ rolls later, Joshua rolls the first $3\textit{ after}$ the first $2$ that he rolled $\textit{after}$ the first $1$ that he rolled. His first $31$ rolls make the sequence \[4,3,11,3,11,8,5,2,12,9,5,7,11,3,6,10,\textbf{1},8,3,\textbf{2},10,4,2,8,1,9,7,12,11,4,\textbf{3}.\] Joshua wonders how many times he should expect to roll the $12$-sided die so that he can remove all but $12$ of the numbers from the entire sequence of rolls and (without changing the order of the sequence), be left with the sequence \[1,2,3,4,5,6,7,8,9,10,11,12.\] What is the expected value of the number of times Joshua must roll the die before he has such a sequence? (Assume Joshua starts from the beginning - do $\textit{not}$ assume he starts by rolling the specific sequence of $31$ rolls above.)

The circuit diagram drawn (see figure ) contains a battery $B$, a lamp $L$ and five switches $S_1$ to $S_5$. The probability that switch $S_3$ is closed (makes contact) is $\frac23$, for the other four switches that probability is $\frac12$ (the probabilities are mutually independent). Calculate the probability that the light is on. [asy] unitsize (2 cm); draw((-1,1)--(-0.5,1)); draw((-0.25,1)--(1,1)--(1,0.25)); draw((1,-0.25)--(1,-1)--(0.05,-1)); draw((-0.05,-1)--(-1,-1)--(-1,0.25)); draw((-1,0.5)--(-1,1)); draw((-1,1)--(-0.5,0.5)); draw((-0.25,0.25)--(0,0)); draw((-1,0)--(-0.75,0)); draw((-0.5,0)--(0,0)); draw((0,1)--(0,0.75)); draw((0,0.5)--(0,0)); draw((-0.25,1)--(-0.5,1.25)); draw((-1,0.25)--(-1.25,0.5)); draw((-0.5,0.5)--(-0.25,0.5)); draw((0,0.75)--(0.25,0.5)); draw((-0.75,0)--(-0.5,-0.25)); draw(Circle((1,0),0.25)); draw(((1,0) + 0.25*dir(45))--((1,0) + 0.25*dir(225))); draw(((1,0) + 0.25*dir(135))--((1,0) + 0.25*dir(315))); draw((0.05,-0.9)--(0.05,-1.1)); draw((-0.05,-0.8)--(-0.05,-1.2)); label("$L$", (1.25,0), E); label("$B$", (-0.1,-1.1), SW); label("$S_1$", (-0.5,1.25), NE); label("$S_2$", (-1.25,0.5), SW); label("$S_3$", (-0.5,0.5), SW); label("$S_4$", (0.25,0.5), NE); label("$S_5$", (-0.5,-0.25), SW); [/asy]

The number $21!=51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd? $\textbf{(A)} \frac{1}{21} \qquad \textbf{(B)} \frac{1}{19} \qquad \textbf{(C)} \frac{1}{18} \qquad \textbf{(D)} \frac{1}{2} \qquad \textbf{(E)} \frac{11}{21}$

Let $S$ be the set of integers which are both a multiple of $70$ and a factor of $630{,}000$. A random element $c$ of $S$ is selected. If the probability that there exists an integer $d$ with $\gcd (c,d) = 70$ and $\operatorname{lcm} (c,d) = 630{,}000$ is $\frac mn$ for some relatively prime integers $m$ and $n$, compute $100m+n$. [i]Proposed by Eugene Chen[/i]

Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn the probability that a player knocks the bottle off the ledge is $\frac{1}{2}$, independently of what has happened before. What is the probability that Larry wins the game? $\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{3}{5}\qquad\textbf{(C) }\frac{2}{3}\qquad\textbf{(D) }\frac{3}{4}\qquad\textbf{(E) }\frac{4}{5}$

Anna writes a sequence of integers starting with the number 12. Each subsequent integer she writes is chosen randomly with equal chance from among the positive divisors of the previous integer (including the possibility of the integer itself). She keeps writing integers until she writes the integer 1 for the first time, and then she stops. One such sequence is \[ 12, 6, 6, 3, 3, 3, 1. \] What is the expected value of the number of terms in Anna’s sequence?

A fair six-sided die is rolled five times. The probability that the five die rolls form an increasing sequence where each value is strictly larger than the one that preceded can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

If I roll three fair $4$-sided dice, what is the probability that the sum of the resulting numbers is relatively prime to the product of the resulting numbers?

Maya lists all the positive divisors of $ 2010^2$. She then randomly selects two distinct divisors from this list. Let $ p$ be the probability that exactly one of the selected divisors is a perfect square. The probability $ p$ can be expressed in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.

Let $ABC$ be an equilateral triangle with each side of length 1. Let $X$ be a point chosen uniformly at random on side $\overline{AB}$. Let $Y$ be a point chosen uniformly at random on side $\overline{AC}$. (Points $X$ and $Y$ are chosen independently.) Let $p$ be the probability that the distance $XY$ is at most $\dfrac{1}{\sqrt[4]{3}}\,$. What is the value of $900p$, rounded to the nearest integer?

Let $n\geq 3$ be integer. Given two pairs of $n$ cards numbered from 1 to $n$. Mix the $2n$ cards up and take the card 3 times every one card. Denote $X_1,\ X_2,\ X_3$ the numbers of the cards taken out in this order taken the cards. Find the probabilty such that $X_1<X_2<X_3$. Note that once a card taken out, it is not taken a back.

In a skiing jump competition $65$ contestants take part. They jump with the previously established order. Each of them jumps once. We assume that the obtained results are different and the orders of the contestants after the competition are equally likely. In each moment of the competition by a leader we call a person who is scored the best at this moment. Denote by $p$ the probability that during the whole competition there was exactly one change of the leader. Prove that $p > 1/16$.

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 $|M-N|$?

Let $X$ be normally distributed with mean $\mu$ and variance $\sigma^2>0$. What is the variance of $e^X$?

An object moves $ 8$ cm in a straight line from $ A$ to $ B$, turns at an angle $ \alpha$, measured in radians and chosen at random from the interval $ (0,\pi)$, and moves $ 5$ cm in a straight line to $ C$. What is the probability that $ AC<7$? $ \textbf{(A)}\ \frac{1}{6} \qquad \textbf{(B)}\ \frac{1}{5} \qquad \textbf{(C)}\ \frac{1}{4} \qquad \textbf{(D)}\ \frac{1}{3} \qquad \textbf{(E)}\ \frac{1}{2}$

Two real numbers are selected independently at random from the interval [-20, 10]. What is the probability that the product of those numbers is greater than zero? $ \textbf{(A)}\ \frac{1}{9} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{4}{9} \qquad \textbf{(D)}\ \frac{5}{9} \qquad \textbf{(E)}\ \frac{2}{3} $