An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.58$. Find $N$.

Suppose $W(k,2)$ is the smallest number such that if $n\ge W(k,2)$, for each coloring of the set $\{1,2,...,n\}$ with two colors there exists a monochromatic arithmetic progression of length $k$. Prove that $W(k,2)=\Omega (2^{\frac{k}{2}})$.

Two unit squares are selected at random without replacement from an $n\times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected squares are horizontally or vertically adjacent is less than $\frac{1}{2015}$.

The problems are really difficult to find online, so here are the problems. P1. It is known that two positive integers $m$ and $n$ satisfy $10n - 9m = 7$ dan $m \leq 2018$. The number $k = 20 - \frac{18m}{n}$ is a fraction in its simplest form. a) Determine the smallest possible value of $k$. b) If the denominator of the smallest value of $k$ is (equal to some number) $N$, determine all positive factors of $N$. c) On taking one factor out of all the mentioned positive factors of $N$ above (specifically in problem b), determine the probability of taking a factor who is a multiple of 4. I added this because my translation is a bit weird. [hide=Indonesian Version] Diketahui dua bilangan bulat positif $m$ dan $n$ dengan $10n - 9m = 7$ dan $m \leq 2018$. Bilangan $k = 20 - \frac{18m}{n}$ merupakan suatu pecahan sederhana. a) Tentukan bilangan $k$ terkecil yang mungkin. b) Jika penyebut bilangan $k$ terkecil tersebut adalah $N$, tentukan semua faktor positif dari $N$. c) Pada pengambilan satu faktor dari faktor-faktor positif $N$ di atas, tentukan peluang terambilnya satu faktor kelipatan 4.[/hide] P2. Let the functions $f, g : \mathbb{R} \to \mathbb{R}$ be given in the following graphs. [hide=Graph Construction Notes]I do not know asymptote, can you please help me draw the graphs? Here are its complete description: For both graphs, draw only the X and Y-axes, do not draw grids. Denote each axis with $X$ or $Y$ depending on which line you are referring to, and on their intercepts, draw a small node (a circle) then mark their $X$ or $Y$ coordinates only (since their other coordinates are definitely 0). Graph (1) is the function $f$, who is a quadratic function with -2 and 4 as its $X$-intercepts and 4 as its $Y$-intercept. You also put $f$ right besides the curve you have, preferably just on the right-up direction of said curve. Graph (2) is the function $g$, which is piecewise. For $x \geq 0$, $g(x) = \frac{1}{2}x - 2$, whereas for $x < 0$, $g(x) = - x - 2$. You also put $g$ right besides the curve you have, on the lower right of the line, on approximately $x = 2$.[/hide] Define the function $g \circ f$ with $(g \circ f)(x) = g(f(x))$ for all $x \in D_f$ where $D_f$ is the domain of $f$. a) Draw the graph of the function $g \circ f$. b) Determine all values of $x$ so that $-\frac{1}{2} \leq (g \circ f)(x) \leq 6$. P3. The quadrilateral $ABCD$ has side lengths $AB = BC = 4\sqrt{3}$ cm and $CD = DA = 4$ cm. All four of its vertices lie on a circle. Calculate the area of quadrilateral $ABCD$. P4. There exists positive integers $x$ and $y$, with $x < 100$ and $y > 9$. It is known that $y = \frac{p}{777} x$, where $p$ is a 3-digit number whose number in its tens place is 5. Determine the number/quantity of all possible values of $y$. P5. The 8-digit number $\overline{abcdefgh}$ (the original problem does not have an overline, which I fixed) is arranged from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$. Such number satisfies $a + c + e + g \geq b + d + f + h$. Determine the quantity of different possible (such) numbers.

A fair $6$-sided die is repeatedly rolled until a $1, 4, 5$, or $6$ is rolled. What is the expected value of the product of all the rolls?

Crisp All, a basketball player, is [i]dropping dimes[/i] and nickels on a number line. Crisp drops a dime on every positive multiple of $10$, and a nickel on every multiple of $5$ that is not a multiple of $10$. Crisp then starts at $0$. Every second, he has a $\frac{2}{3}$ chance of jumping from his current location $x$ to $x+3$, and a $\frac{1}{3}$ chance of jumping from his current location $x$ to $x+7$. When Crisp jumps on either a dime or a nickel, he stops jumping. What is the probability that Crisp [i]stops on a dime[/i]?

An unfair coin lands on heads with a probability of $\tfrac{1}{4}$. When tossed $n$ times, the probability of exactly two heads is the same as the probability of exactly three heads. What is the value of $n$? $ \textbf {(A) } 5 \qquad \textbf {(B) } 8 \qquad \textbf {(C) } 10 \qquad \textbf {(D) } 11 \qquad \textbf {(E) } 13 $

Alice and Bob find themselves on a coordinate plane at time $t=0$ at $A(1,0)$ and $B(-1,0)$ respectively. They have no sense of direction, but they want to find each other. They each pick a direction independently and with uniform random probability. Both Alice and Bob travel at a constant speed of $1 \frac{unit}{min}$ in their chosen directions. They continue on their straight line paths forever, each hoping to catch sight of the other. They both have a $1$ unit radius of view; they can see something if and only if its distance from them is at most $1$ unit. What is the probability they never see each other?

The numbers $1,2,\dots,9$ are randomly placed into the $9$ squares of a $3 \times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd? $\textbf{(A) }1/21\qquad\textbf{(B) }1/14\qquad\textbf{(C) }5/63\qquad\textbf{(D) }2/21\qquad\textbf{(E) } 1/7$

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]

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 $n$ and $k$ be positive integers with $1 \leq k \leq n$. A set of cards numbered $1$ to $n$ are arranged randomly in a row from left to right. A person alternates between performing the following moves: [list=a] [*] The leftmost card in the row is moved $k-1$ positions to the right while the cards in positions $2$ through $k$ are each moved one place to the left. [*] The rightmost card in the row is moved $k-1$ positions to the left while the cards in positions $n-k+1$ through $n-1$ are each moved one place to the right. [/list] Determine the probability that after some number of moves the cards end up in order from $1$ to $n$, left to right.

A bee is moving in three-dimensional space. A fair six-sided die with faces labeled $A^+, A^-, B^+, B^-, C^+$, and $C^-$ is rolled. Suppose the bee occupies the point $(a, b, c)$. If the die shows $A^+$, then the bee moves to the point $(a+1, b, c)$ and if the die shows $A^-$, then the bee moves to the point $(a-1, b, c)$. Analogous moves are made with the other four outcomes. Suppose the bee starts at the point $(0, 0, 0)$ and the die is rolled four times. What is the probability that the bee traverses four distinct edges of some unit cube? $ \textbf{(A) }\frac{1}{54} \qquad \textbf{(B) }\frac{7}{54} \qquad \textbf{(C) }\frac{1}{6} \qquad \textbf{(D) }\frac{5}{18} \qquad \textbf{(E) }\frac{2}{5} \qquad $

A hotel packed breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls and once wrapped, the rolls were indistinguishable from one another. She then randomly put three rolls in a bag for each of the guests. Given that the probability each guest got one roll of each type is $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m+n$.

At the Rice Mathematics Tournament, 80% of contestants wear blue jeans, 70% wear tennis shoes, and 80% of those who wear blue jeans also wear tennis shoes. What fraction of people wearing tennis shoes are wearing blue jeans?

Objects $A$ and $B$ move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object $A$ starts at $(0,0)$ and each of its steps is either right or up, both equally likely. Object $B$ starts at $(5,7)$ and each of its steps is either left or down, both equally likely. Which of the following is closest to the probability that the objects meet? $ \textbf{(A)}\ 0.10 \qquad \textbf{(B)}\ 0.15 \qquad \textbf{(C)}\ 0.20 \qquad \textbf{(D)}\ 0.25 \qquad \textbf{(E)}\ 0.30$

One day while the Kubik family attends one of Michael's baseball games, Tony gets bored and walks to the creek a few yards behind the baseball field. One of Tony's classmates Mitchell sees Tony and goes to join him. While playing around the creek, the two boys find an ordinary six-sided die buried in sediment. Mitchell washes it off in the water and challenges Tony to a contest. Each of the boys rolls the die exactly once. Mitchell's roll is $3$ higher than Tony's. "Let's play once more," says Tony. Let $a/b$ be the probability that the difference between the outcomes of the two dice is again exactly $3$ (regardless of which of the boys rolls higher), where $a$ and $b$ are relatively prime positive integers. Find $a+b$.

The figure on the right shows the map of Squareville, where each city block is of the same length. Two friends, Alexandra and Brianna, live at the corners marked by $A$ and $B$, respectively. They start walking toward each other's house, leaving at the same time, walking with the same speed, and independently choosing a path to the other's house with uniform distribution out of all possible minimum-distance paths [that is, all minimum-distance paths are equally likely]. What is the probability they will meet? [asy] size(200); defaultpen(linewidth(0.8)); for(int i=0;i<=2;++i) { for(int j=0;j<=4;++j) { draw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--cycle); } } for(int i=3;i<=4;++i) { for(int j=3;j<=6;++j) { draw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--cycle); } } label("$A$",origin,SW); label("$B$",(5,7),SE); [/asy]

Three numbers are selected at random from the interval $[0,1]$. What is the probability that they form the lengths of the sides of a triangle?

Prove that there exists a positive $ c$ such that for every positive integer $ N$ among any $ N$ positive integers not exceeding $ 2N$ there are two numbers whose greatest common divisor is greater than $ cN$.

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.

Twenty-seven players are randomly split into three teams of nine. Given that Zack is on a different team from Mihir and Mihir is on a different team from Andrew, what is the probability that Zack and Andrew are on the same team?

A point in the plane, both of whose rectangular coordinates are integers with absolute values less than or equal to four, is chosen at random, with all such points having an equal probability of being chosen. What is the probability that the distance from the point to the origin is at most two units? $\textbf{(A) }\frac{13}{81}\qquad\textbf{(B) }\frac{15}{81}\qquad\textbf{(C) }\frac{13}{64}\qquad\textbf{(D) }\frac{\pi}{16}\qquad \textbf{(E) }\text{the square of a rational number}$

Let $ \vartheta_1,...,\vartheta_n$ be independent, uniformly distributed, random variables in the unit interval $ [0,1]$. Define \[ h(x)\equal{} \frac1n \# \{k: \; \vartheta_k<x\ \}.\] Prove that the probability that there is an $ x_0 \in (0,1)$ such that $ h(x_0)\equal{}x_0$, is equal to $ 1\minus{} \frac1n.$ [i]G. Tusnady[/i]

Crisp All, a basketball player, is [i]dropping dimes[/i] and nickels on a number line. Crisp drops a dime on every positive multiple of $10$, and a nickel on every multiple of $5$ that is not a multiple of $10$. Crisp then starts at $0$. Every second, he has a $\frac{2}{3}$ chance of jumping from his current location $x$ to $x+3$, and a $\frac{1}{3}$ chance of jumping from his current location $x$ to $x+7$. When Crisp jumps on either a dime or a nickel, he stops jumping. What is the probability that Crisp [i]stops on a dime[/i]?