Take the square with vertices $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. Choose a random point in this square and draw the line segment from it to $(0,0)$. Choose a second random point in this square and draw the line segment from it to $(1,0)$. What is the probability that the two line segments intersect?

A game about passing barriers rules that in the $n$th barrier, you need to throw a dice for $n$ times. If the sum of points you get is larger than $2^n$, then you can pass this barrier. [b](a)[/b] How many barriers can you pass at most? [b](b)[/b] Find the probablity of passing the first three barriers.

Consider a coin with a head toss probability $p$ where $0 <p <1$ is fixed. Toss the coin several times, the tosses should be independent of each other. Denote by $A_i$ the event that of the $i$-th, $(i + 1)$-th, $\ldots$ , the $(i+m-1)$-th throws, exactly $T$ is the tail. For $T = 1$, calculate the conditional probability $\mathbb{P}(\bar{A_2} \bar{A_3} \cdots \bar{A_m} | A_1)$, and for $T = 2$, prove that $\mathbb{P}(\bar{A_2} \bar{A_3} \cdots \bar{A_m} | A_1)$ has approximation in the form $a+ \tfrac{b}{m} + \mathcal{O}(p^m)$ as $m \to \infty$.

Reimu and Sanae play a game using $4$ fair coins. Initially both sides of each coin are white. Starting with Reimu, they take turns to color one of the white sides either red or green. After all sides are colored, the four coins are tossed. If there are more red sides showing up, then Reimu wins, and if there are more green sides showing up, then Sanae wins. However, if there is an equal number of red sides and green sides, then [i]neither[/i] of them wins. Given that both of them play optimally to maximize the probability of winning, what is the probability that Reimu wins?

In the game of "Fingers", $N$ players stand in a circle and simultaneously thrust out their right hands, each with a certain number of fingers showing. The total number of fingers shown is counted out round the circle from the leader, and the player on whom the count stops is the winner. How large must $N$ be for a suitably chosen group of $N/10$ players to contain a winner with probability at least $0.9$? How does the probability that the leader wins behave as $N\to\infty$?

Let $P$ be an arbitrary point inside $\triangle ABC$ with sides $3,7,8$. What is the probability that the distance of $P$ to at least one vertices of the triangle is less than $1$? $ \textbf{(A)}\ \frac{\pi}{36}\sqrt 2 \qquad\textbf{(B)}\ \frac{\pi}{36}\sqrt 3 \qquad\textbf{(C)}\ \frac{\pi}{36} \qquad\textbf{(D)}\ \frac12 \qquad\textbf{(E)}\ \frac 34 $

Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is $25.$ One marble is taken out of each box randomly. The probability that both marbles are black is $27/50,$ and the probability that both marbles are white is $m/n,$ where $m$ and $n$ are relatively prime positive integers. What is $m+n?$

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?

Let $ S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $ P$ whose vertices are in $ S$, let $ a(P)$ be the number of vertices of $ P$, and let $ b(P)$ be the number of points of $ S$ which are outside $ P$. A line segment, a point, and the empty set are considered as convex polygons of $ 2$, $ 1$, and $ 0$ vertices respectively. Prove that for every real number $ x$ \[\sum_{P}{x^{a(P)}(1 \minus{} x)^{b(P)}} \equal{} 1,\] where the sum is taken over all convex polygons with vertices in $ S$. [i]Alternative formulation[/i]: Let $ M$ be a finite point set in the plane and no three points are collinear. A subset $ A$ of $ M$ will be called round if its elements is the set of vertices of a convex $ A \minus{}$gon $ V(A).$ For each round subset let $ r(A)$ be the number of points from $ M$ which are exterior from the convex $ A \minus{}$gon $ V(A).$ Subsets with $ 0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $ A$ of $ M$ construct the polynomial \[ P_A(x) \equal{} x^{|A|}(1 \minus{} x)^{r(A)}. \] Show that the sum of polynomials for all round subsets is exactly the polynomial $ P(x) \equal{} 1.$ [i]Proposed by Federico Ardila, Colombia[/i]

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 $m+n$.

Two different squares are randomly chosen from an $8\times8$ chessboard. What is the probability that two queens placed on the two squares can attack each other? Recall that queens in chess can attack any square in a straight line vertically, horizontally, or diagonally from their current position.

Al, Bill, and Cal will each randomly be assigned a whole number from $1$ to $10$, inclusive, with no two of them getting the same number. What is the probability that Al's number will be a whole number multiple of Bill's and Bill's number will be a whole number multiple of Cal's? $\textbf{(A) } \dfrac{9}{1000} \qquad\textbf{(B) } \dfrac{1}{90} \qquad\textbf{(C) } \dfrac{1}{80} \qquad\textbf{(D) } \dfrac{1}{72} \qquad\textbf{(E) } \dfrac{2}{121} $

Prinstan Trollner and Dukejukem are competing at the game show WASS. Both players spin a wheel which chooses an integer from $1$ to $50$ uniformly at random, and this number becomes their score. Dukejukem then flips a weighted coin that lands heads with probability $\tfrac{3}{5}$. If he flips heads, he adds $1$ to his score. A player wins the game if their score is higher than the other player's score. A player wins the game if their score is higher than the other player's score. The probability Dukejukem defeats the Trollner to win WASS equals $\tfrac{m}{n}$ where $m$ and $n$ are coprime positive integers. Computer $m+n$.

We select a real number $\alpha$ uniformly and at random from the interval $(0,500)$. Define \[ S = \frac{1}{\alpha} \sum_{m=1}^{1000} \sum_{n=m}^{1000} \left\lfloor \frac{m+\alpha}{n} \right\rfloor. \] Let $p$ denote the probability that $S \ge 1200$. Compute $1000p$. [i]Proposed by Evan Chen[/i]

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 $a+b$.

At Rachelle's school an A counts 4 points, a B 3 points, a C 2 points, and a D 1 point. Her GPA on the four classes she is taking is computed as the total sum of points divided by $4$. She is certain that she will get As in both Mathematics and Science, and at least a C in each of English and History. She think she has a $\frac{1}{6}$ chance of getting an A in English, and a $\frac{1}{4}$ chance of getting a B. In History, she has a $\frac{1}{4}$ chance of getting an A, and a $\frac{1}{3}$ chance of getting a B, independently of what she gets in English. What is the probability that Rachelle will get a GPA of at least 3.5? $\textbf{(A) }\frac{11}{72}\qquad\textbf{(B) }\frac{1}{6}\qquad\textbf{(C) }\frac{3}{16}\qquad\textbf{(D) }\frac{11}{24}\qquad\textbf{(E) }\frac{1}{2}$

In a moment of impaired thought, Joe decides he wants to dress up as a member of NSYNC for his school Halloween party that night. If he dresses up as JC Chasez, he has a probability of $25\%$ of getting beat up at the party. If he dresses up as Justin Timberlake, he has a $60\%$ probability of getting beat up at the party. If he dresses up as any other member of NSYNC, he won’t get beat up because no one will recognize his costume. If there is an equal probability of him dressing up as any of the $5$ NSYNC members, what is the probability he will get beat up at the Halloween party?

Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime? $ \textbf{(A)}\ \left(\frac{1}{12}\right)^{12}\qquad \textbf{(B)}\ \left(\frac{1}{6}\right)^{12}\qquad \textbf{(C)}\ 2\left(\frac{1}{6}\right)^{11}\qquad \textbf{(D)}\ \frac{5}{2}\left(\frac{1}{6}\right)^{11}\qquad \textbf{(E)}\ \left(\frac{1}{6}\right)^{10}$

Four dice are thrown. What is the probability that the product of the number equals $ 36?$

$n\in{Z^{+}}$ and $A={1,\ldots ,n}$. $f: N\rightarrow N$ and $\sigma: N\rightarrow N$ are two permutations, if there is one $k\in A$ such that $(f\circ\sigma)(1),\ldots ,(f\circ\sigma)(k)$ is increasing and $(f\circ\sigma)(k),\ldots ,(f\circ\sigma)(n)$ is decreasing sequences we say that $f$ is good for $\sigma$. $S_\sigma$ shows the set of good functions for $\sigma$. a) Prove that, $S_\sigma$ has got $2^{n-1}$ elements for every $\sigma$ permutation. b)$n\geq 4$, prove that there are permutations $\sigma$ and $\tau$ such that, $S_{\sigma}\cap S_{\tau}=\phi$ .

A deck of forty cards consists of four 1's, four 2's,..., and four 10's. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let $m/n$ be the probability that two randomly selected cards also form a pair, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Among $100$ points in the plane, no three collinear, exactly $4026$ pairs are connected by line segments. Each point is then randomly assigned an integer from $1$ to $100$ inclusive, each equally likely, such that no integer appears more than once. Find the expected value of the number of segments which join two points whose labels differ by at least $50$. [i]Proposed by Evan Chen[/i]

Tom is searching for the $6$ books he needs in a random pile of $30$ books. What is the expected number of books must he examine before finding all $6$ books he needs?

Let $ a_1, a_2, \ldots , a_n$ be distinct positive integers and let $ M$ be a set of $ n \minus{} 1$ positive integers not containing $ s \equal{} a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n.$ A grasshopper is to jump along the real axis, starting at the point $ 0$ and making $ n$ jumps to the right with lengths $ a_1, a_2, \ldots , a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $ M.$ [i]Proposed by Dmitry Khramtsov, Russia[/i]

1. What is the probability that a randomly chosen word of this sentence has exactly four letters?