Aaron takes a square sheet of paper, with one corner labeled $A$. Point $P$ is chosen at random inside of the square and Aaron folds the paper so that points $A$ and $P$ coincide. He cuts the sheet along the crease and discards the piece containing $A$. Let $p$ be the probability that the remaining piece is a pentagon. Find the integer nearest to $100p$. [i]Proposed by Aaron Lin[/i]

The George Washington Bridge is $2016$ meters long. Sally is standing on the George Washington Bridge, $1010$ meters from its left end. Each step, she either moves $1$ meter to the left or $1$ meter to the right, each with probability $\dfrac{1}{2}$. What is the expected number of steps she will take to reach an end of the bridge?

A parking lot has $ 16$ spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers chose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires $ 2$ adjacent spaces. What is the probability that she is able to park? $ \textbf{(A)} \ \frac {11}{20} \qquad \textbf{(B)} \ \frac {4}{7} \qquad \textbf{(C)} \ \frac {81}{140} \qquad \textbf{(D)} \ \frac {3}{5} \qquad \textbf{(E)} \ \frac {17}{28}$

At a party there are only single women and married men with their wives. The probability that a randomly selected woman is single is $\frac{2}{5}$. What fraction of the people in the room are married men? $\textbf{(A)}\ \frac{1}{3}\qquad \textbf{(B)}\ \frac{3}{8}\qquad \textbf{(C)}\ \frac{2}{5}\qquad \textbf{(D)}\ \frac{5}{12}\qquad \textbf{(E)}\ \frac{3}{5}$

An envelope contains eight bills: $ 2$ ones, $ 2$ fives, $ 2$ tens, and $ 2$ twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $ \$ 20$ or more? $ \textbf{(A)}\ \frac {1}{4}\qquad \textbf{(B)}\ \frac {2}{7}\qquad \textbf{(C)}\ \frac {3}{7}\qquad \textbf{(D)}\ \frac {1}{2}\qquad \textbf{(E)}\ \frac {2}{3}$

A [i]triangulation[/i] of a polygon is a subdivision of the polygon into triangles meeting edge to edge, with the property that the set of triangle vertices coincides with the set of vertices of the polygon. Adam randomly selects a triangulation of a regular $180$-gon. Then, Bob selects one of the $178$ triangles in this triangulation. The expected number of $1^\circ$ angles in this triangle can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$. [i]Proposed by Lewis Chen[/i]

Three persons $A,B,C$, are playing the following game: A $k$-element subset of the set $\{1, . . . , 1986\}$ is randomly chosen, with an equal probability of each choice, where $k$ is a fixed positive integer less than or equal to $1986$. The winner is $A,B$ or $C$, respectively, if the sum of the chosen numbers leaves a remainder of $0, 1$, or $2$ when divided by $3$. For what values of $k$ is this game a fair one? (A game is fair if the three outcomes are equally probable.)

Let $F_n$ be the set of all bijective functions $f:\left\{1,2,\ldots,n\right\}\rightarrow\left\{1,2,\ldots,n\right\}$ that satisfy the conditions: 1. $f(k)\leq k+1$ for all $k\in\left\{1,2,\ldots,n\right\}$ 2. $f(k)\neq k$ for all $k\in\left\{2,3,\ldots,n\right\}$ Find the probability that $f(1)\neq1$ for $f$ randomly chosen from $F_n$.

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.

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}$

Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of its opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube? $ \textbf{(A)}\ \frac {1}{8}\qquad \textbf{(B)}\ \frac {3}{16}\qquad \textbf{(C)}\ \frac {1}{4} \qquad \textbf{(D)}\ \frac {3}{8}\qquad \textbf{(E)}\ \frac {1}{2}$

In a culturing of bacteria, there are two species of them: red and blue bacteria. When two red bacteria meet, they transform into one blue bacterium. When two blue bacteria meet, they transform into four red bacteria. When a red and a blue bacteria meet, they transform into three red bacteria. Find, in function of the amount of blue bacteria and the red bacteria initially in the culturing, all possible amounts of bacteria, and for every possible amount, the possible amounts of red and blue bacteria.

A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is probability that the dart lands within the center square? [asy] unitsize(10mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(0,1), B=(1,0), C=(1+sqrt(2),0), D=(2+sqrt(2),1), E=(2+sqrt(2),1+sqrt(2)), F=(1+sqrt(2),2+sqrt(2)), G=(1,2+sqrt(2)), H=(0,1+sqrt(2)); draw(A--B--C--D--E--F--G--H--cycle); draw(A--D); draw(B--G); draw(C--F); draw(E--H); [/asy] $ \textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad\textbf{(E)}\ 2 - \sqrt{2}$

A box contains $p$ white balls and $q$ black balls. Beside the box there is a pile of black balls. Two balls are taken out of the box. If they have the same color, a black ball from the pile is put into the box. If they have different colors, the white ball is put back into the box. This procedure is repeated until the last two balls are removed from the box and one last ball is put in. What is the probability that this last ball is white?

Let $K{}$ be an equilateral triangle of unit area, and choose $n{}$ independent random points uniformly from $K{}$. Let $K_n$ be the intersection of all translations of $K{}$ that contain all the selected points. Determine the expected value of the area of $K_n.$

Ankit, Box, and Clark are taking the tiebreakers for the geometry round, consisting of three problems. Problem $k$ takes each $k$ minutes to solve. If for any given problem there is a $\frac13$ chance for each contestant to solve that problem first, what is the probability that Ankit solves a problem first?

A point $ (x,y)$ is randomly picked from inside the rectangle with vertices $ (0,0)$, $ (4,0)$, $ (4,1)$, and $ (0,1)$. What is the probability that $ x<y$? $ \textbf{(A)}\ \frac{1}{8} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{3}{8} \qquad \textbf{(D)}\ \frac{1}{2} \qquad \textbf{(E)}\ \frac{3}{4}$

Three numbers, $a_1$, $a_2$, $a_3$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 1000\}$. Three other numbers, $b_1$, $b_2$, $b_3$, are then drawn randomly and without replacement from the remaining set of 997 numbers. Let $p$ be the probability that, after a suitable rotation, a brick of dimensions $a_1 \times a_2 \times a_3$ can be enclosed in a box of dimensions $b_1 \times b_2 \times b_3$, with the sides of the brick parallel to the sides of the box. If $p$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?

There are 10 cards, labeled from 1 to 10. Three cards denoted by $ a,\ b,\ c\ (a > b > c)$ are drawn from the cards at the same time. Find the probability such that $ \int_0^a (x^2 \minus{} 2bx \plus{} 3c)\ dx \equal{} 0$.

Aws plays a solitaire game on a fifty-two card deck: whenever two cards of the same color are adjacent, he can remove them. Aws wins the game if he removes all the cards. If Aws starts with the cards in a random order, what is the probability for him to win?

Yoknapatawpha County has $500,000$ families. Each family is expected to continue to have children until it has a girl, at which point each family stops having children. If the probability of having a boy is $50\%$, and no families have either fertility problems or multiple children per birthing, how many families are expected to have at least $5$ children?

Two real numbers between $0$ and $1$ are randomly chosen. Calculate the probability that any one of them is less than the square of the other.

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}$

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.$

Debra flips a fair coin repeatedly, keeping track of how many heads and how many tails she has seen in total, until she gets either two heads in a row or two tails in a row, at which point she stops flipping. What is the probability that she gets two heads in a row but she sees a second tail before she sees a second head? $\textbf{(A) } \frac{1}{36} \qquad \textbf{(B) } \frac{1}{24} \qquad \textbf{(C) } \frac{1}{18} \qquad \textbf{(D) } \frac{1}{12} \qquad \textbf{(E) } \frac{1}{6}$