There is a board numbered $-10$ to $10$. Each square is coloured either red or white, and the sum of the numbers on the red squares is $n$. Maureen starts with a token on the square labeled $0$. She then tosses a fair coin ten times. Every time she flips heads, she moves the token one square to the right. Every time she flips tails, she moves the token one square to the left. At the end of the ten flips, the probability that the token finishes on a red square is a rational number of the form $\frac a b$. Given that $a + b = 2001$, determine the largest possible value for $n$.

A dog has three trainers: [list] [*]The first trainer gives him a treat right away. [*]The second trainer makes him jump five times, then gives him a treat. [*]The third trainer makes him jump three times, then gives him no treat. [/list] The dog will keep picking trainers with equal probability until he gets a treat. (The dog's memory isn't so good, so he might pick the third trainer repeatedly!) What is the expected number of times the dog will jump before getting a treat?

Five girls and five boys randomly sit in ten seats that are equally spaced around a circle. The probability that there is at least one diameter of the circle with two girls sitting on opposite ends of the diameter is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

In chess, there are two types of minor pieces, the bishop and the knight. A bishop may move along a diagonal, as long as there are no pieces obstructing its path. A knight may jump to any lattice square $\sqrt{5}$ away as long as it isn't occupied. One day, a bishop and a knight were on squares in the same row of an infinite chessboard, when a huge meteor storm occurred, placing a meteor in each square on the chessboard independently and randomly with probability $p$. Neither the bishop nor the knight were hit, but their movement may have been obstructed by the meteors. The value of $p$ that would make the expected number of valid squares that the bishop can move to and the number of squares that the knight can move to equal can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$. [i]Proposed by Lewis Chen[/i]

We randomly choose 5 distinct positive integers less than or equal to 90. What is the floor of 10 times the expected value of the fourth largest number?

Let $A$ be the set of positive integer divisors of $2025$. Let $B$ be a randomly selected subset of $A$. The probability that $B$ is a nonempty set with the property that the least common multiple of its element is $2025$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Starting at $(0, 0)$, Richard takes $2n+1$ steps, with each step being one unit either East, North, West, or South. For each step, the direction is chosen uniformly at random from the four possibilities. Determine the probability that Richard ends at $(1, 0)$.

Define an ordered quadruple of integers $(a, b, c, d)$ as interesting if $1 \le a<b<c<d \le 10$, and $a+d>b+c$. How many ordered quadruples are there?

Let $T=\text{TNFTPP}$. Fermi and Feynman play the game $\textit{Probabicloneme}$ in which Fermi wins with probability $a/b$, where $a$ and $b$ are relatively prime positive integers such that $a/b<1/2$. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play $\textit{Probabicloneme}$ so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is $(T-332)/(2T-601)$. Find the value of $a$.

Nine balls numbered $1,2,\cdots,9$ are put on nine poines that divide the circle into nine equal parts. The sum of absolute values of the difference between the number of two adjacent balls is $S$. Find the probablity of $S$ takes its minumum value. Note: If one way of putting balls can be the same as another one by rotating or specular-reflecting, then they are considered the same way.

Bernardo randomly picks $ 3$ distinct numbers from the set $ \{1,2,3,4,5,6,7,8,9\}$ and arranges them in descending order to form a $ 3$-digit number. Silvia randomly picks $ 3$ distinct numbers from the set $ \{1,2,3,4,5,6,7,8\}$ and also arranges them in descending order to form a $ 3$-digit number. What is the probability that Bernardo's number is larger than Silvia's number? $ \textbf{(A)}\ \frac {47}{72}\qquad \textbf{(B)}\ \frac {37}{56}\qquad \textbf{(C)}\ \frac {2}{3}\qquad \textbf{(D)}\ \frac {49}{72}\qquad \textbf{(E)}\ \frac {39}{56}$

On a one-way street, an unending sequence of cars of width $a$, length $b$ passes with velocity $v$. The cars are separated by the distance $c$. A pedestrian crosses the street perpendicularly with velocity $w$, without paying attention to the cars. [b](a)[/b] What is the probability that the pedestrian crosses the street uninjured? [b](b)[/b] Can he improve this probability by crossing the road in a direction other than perpendicular?

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$.

Kelvin the Frog is going to roll three fair ten-sided dice with faces labelled $0, 1, \dots, 9$. First he rolls two dice, and finds the sum of the two rolls. Then he rolls the third die. What is the probability that the sum of the first two rolls equals the third roll?

A bag of popping corn contains $\frac{2}{3}$ white kernels and $\frac{1}{3}$ yellow kernels. Only $\frac{1}{2}$ of the white kernels will pop, whereas $\frac{2}{3}$ of the yellow ones will pop. A kernel is selected at random from the bag, and pops when placed in the popper. What is the probability that the kernel selected was white? $ \textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{5}{9} \qquad\textbf{(C)}\ \frac{4}{7} \qquad\textbf{(D)}\ \frac{3}{5} \qquad\textbf{(E)}\ \frac{2}{3} $

A pair of 8-sided dice have sides numbered 1 through 8. Each side has the same probability (chance) of landing face up. The probability that the product of the two numbers that land face-up exceeds 36 is $\textbf{(A)}\ \dfrac{5}{32} \qquad \textbf{(B)}\ \dfrac{11}{64} \qquad \textbf{(C)}\ \dfrac{3}{16} \qquad \textbf{(D)}\ \dfrac{1}{4} \qquad \textbf{(E)}\ \dfrac{1}{2}$

Jeff rotates spinners $ P$, $ Q$ and $ R$ and adds the resulting numbers. What is the probability that his sum is an odd number? [asy]size(200); path circle=circle((0,0),2); path r=(0,0)--(0,2); draw(circle,linewidth(1)); draw(shift(5,0)*circle,linewidth(1)); draw(shift(10,0)*circle,linewidth(1)); draw(r,linewidth(1)); draw(rotate(120)*r,linewidth(1)); draw(rotate(240)*r,linewidth(1)); draw(shift(5,0)*r,linewidth(1)); draw(shift(5,0)*rotate(90)*r,linewidth(1)); draw(shift(5,0)*rotate(180)*r,linewidth(1)); draw(shift(5,0)*rotate(270)*r,linewidth(1)); draw(shift(10,0)*r,linewidth(1)); draw(shift(10,0)*rotate(60)*r,linewidth(1)); draw(shift(10,0)*rotate(120)*r,linewidth(1)); draw(shift(10,0)*rotate(180)*r,linewidth(1)); draw(shift(10,0)*rotate(240)*r,linewidth(1)); draw(shift(10,0)*rotate(300)*r,linewidth(1)); label("$P$", (-2,2)); label("$Q$", shift(5,0)*(-2,2)); label("$R$", shift(10,0)*(-2,2)); label("$1$", (-1,sqrt(2)/2)); label("$2$", (1,sqrt(2)/2)); label("$3$", (0,-1)); label("$2$", shift(5,0)*(-sqrt(2)/2,sqrt(2)/2)); label("$4$", shift(5,0)*(sqrt(2)/2,sqrt(2)/2)); label("$6$", shift(5,0)*(sqrt(2)/2,-sqrt(2)/2)); label("$8$", shift(5,0)*(-sqrt(2)/2,-sqrt(2)/2)); label("$1$", shift(10,0)*(-0.5,1)); label("$3$", shift(10,0)*(0.5,1)); label("$5$", shift(10,0)*(1,0)); label("$7$", shift(10,0)*(0.5,-1)); label("$9$", shift(10,0)*(-0.5,-1)); label("$11$", shift(10,0)*(-1,0));[/asy] $ \textbf{(A)}\ \dfrac{1}{4} \qquad \textbf{(B)}\ \dfrac{1}{3} \qquad \textbf{(C)}\ \dfrac{1}{2} \qquad \textbf{(D)}\ \dfrac{2}{3} \qquad \textbf{(E)}\ \dfrac{3}{4}$

There are three balls of distinct colors in a bag. We repeatedly draw out the balls one by one, the balls are put back into the bag after each drawing. What is the probability that, after $n$ drawings, (a) exactly one color occured? (b) exactly two oclors occured? (c) all three colors occured?

Everyday at school, Jo climbs a flight of $6$ stairs. Joe can take the stairs $1,2$, or $3$ at a time. For example, Jo could climb $3$, then $1$, then $2$. In how many ways can Jo climb the stairs? $ \textbf{(A)}\ 13 \qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 24 $

For a particular peculiar pair of dice, the probabilities of rolling 1, 2, 3, 4, 5 and 6 on each die are in the ratio $ 1: 2: 3: 4: 5: 6$. What is the probability of rolling a total of 7 on the two dice? $ \textbf{(A) } \frac 4{63} \qquad \textbf{(B) } \frac 18 \qquad \textbf{(C) } \frac 8{63} \qquad \textbf{(D) } \frac 16 \qquad \textbf{(E) } \frac 27$

A traffic light runs repeatedly through the following cycle: green for $ 30$ seconds, then yellow for $ 3$ seconds, and then red for $ 30$ seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching? $ \textbf{(A)}\ \frac {1}{63}\qquad \textbf{(B)}\ \frac {1}{21}\qquad \textbf{(C)}\ \frac {1}{10}\qquad \textbf{(D)}\ \frac {1}{7}\qquad \textbf{(E)}\ \frac {1}{3}$

Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\tfrac12$ is $\tfrac{a-b\pi}c$, where $a$, $b$, and $c$ are positive integers and $\gcd(a,b,c)=1$. What is $a+b+c$? $\textbf{(A) }59\qquad\textbf{(B) }60\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$

DeAndre Jordan shoots free throws that are worth $1$ point each. He makes $40\%$ of his shots. If he takes two shots find the probability that he scores at least $1$ point.

Let $X_1,X_2,\ldots$ be independent and identically distributed random variables with distribution $\mathbb{P}(X_1=0)=\mathbb{P}(X_1=1)=\frac12$. Let $Y_1$, $Y_2$, $Y_3$, and $Y_4$ be independent, identically distributed random variables, where $Y_1:=\sum_{k=1}^\infty \frac{X_k}{16^k}$. Decide whether the random variables $Y_1+2Y_2+4Y_3+8Y_4$ and $Y_1+4Y_3$ are absolutely continuous.

Team $ A$ and team $ B$ play a series. The first team to win three games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team $ B$ wins the second game and team $ A$ wins the series, what is the probability that team $ B$ wins the first game? $ \textbf{(A)}\ \frac{1}{5}\qquad \textbf{(B)}\ \frac{1}{4}\qquad \textbf{(C)}\ \frac{1}{3}\qquad \textbf{(D)}\ \frac{1}{2}\qquad \textbf{(E)}\ \frac{2}{3}$