Four standard six-sided dice are rolled. Find the probability that, for each pair of dice, the product of the two numbers rolled on those dice is a multiple of 4.

I give you a deck of $n$ cards numbered $1$ through $n$. On each turn, you take the top card of the deck and place it anywhere you choose in the deck. You must arrange the cards in numerical order, with card $1$ on top and card $n$ on the bottom. If I place the deck in a random order before giving it to you, and you know the initial order of the cards, what is the expected value of the minimum number of turns you need to arrange the deck in order?

Do there exist such a triangle $T$, that for any coloring of a plane in two colors one may found a triangle $T'$, equal to $T$, such that all vertices of $T'$ have the same color. [b] proposed by S. Berlov[/b]

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$

Given positive integers $m$ and $n$, prove that there is a positive integer $c$ such that the numbers $cm$ and $cn$ have the same number of occurrences of each non-zero digit when written in base ten.

An integer $x$ is selected at random between 1 and $2011!$ inclusive. The probability that $x^x - 1$ is divisible by $2011$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$. [i]Author: Alex Zhu[/i]

The following diagram shows four vertices connected by six edges. Suppose that each of the edges is randomly painted either red, white, or blue. The probability that the three edges adjacent to at least one vertex are colored with all three colors is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/6/4/de0a2a1a659011a30de1859052284c696822bb.png[/img]

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]

Show that reducing the dimensions of a cuboid we can't get another cuboid with half the volume and half the surface.

Given are natural numbers $n>m>1$. We draw $m$ numbers from the set $\{1,2,...,n\}$ one by one without putting the drawn numbers back. Find the expected value of the difference between the largest and the smallest of the drawn numbers.

Rachel has the number $1000$ in her hands. When she puts the number $x$ in her left pocket, the number changes to $x+1.$ When she puts the number $x$ in her right pocket, the number changes to $x^{-1}.$ Each minute, she flips a fair coin. If it lands heads, she puts the number into her left pocket, and if it lands tails, she puts it into her right pocket. She then takes the new number out of her pocket. If the expected value of the number in Rachel's hands after eight minutes is $E,$ compute $\left\lfloor\frac{E}{10}\right\rfloor.$

Jane has two bags $X$ and $Y$. Bag $X$ contains 4 red marbles and 5 blue marbles (and nothing else). Bag $Y$ contains 7 red marbles and 6 blue marbles (and nothing else). Jane will choose one of her bags at random (each bag being equally likely). From her chosen bag, she will then select one of the marbles at random (each marble in that bag being equally likely). What is the probability that she will select a red marble?

The four faces of a tetrahedral die are labelled $0, 1, 2,$ and $3,$ and the die has the property that, when it is rolled, the die promptly vanishes, and a number of copies of itself appear equal to the number on the face the die landed on. For example, if it lands on the face labelled $0,$ it disappears. If it lands on the face labelled $1,$ nothing happens. If it lands on the face labelled $2$ or $3,$ there will then be $2$ or $3$ copies of the die, respectively (including the original). Suppose the die and all its copies are continually rolled, and let $p$ be the probability that they will all eventually disappear. Find $\left\lfloor \frac{10}{p} \right\rfloor$.

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} $

Assume that the number of offspring for every man can be $0,1,\ldots, n$ with with probabilities $p_0,p_1,\ldots,p_n$ independently from each other, where $p_0+p_1+\cdots+p_n=1$ and $p_n\neq 0$. (This is the so-called Galton-Watson process.) Which positive integer $n$ and probabilities $p_0,p_1,\ldots,p_n$ will maximize the probability that the offspring of a given man go extinct in exactly the tenth generation?

A player chooses one of the numbers $ 1$ through $ 4$. After the choice has been made, two regular four-sided (tetrahedral) dice are rolled, with the sides of the dice numbered $ 1$ through $ 4$. If the number chosen appears on the bottom of exactly one die after it is rolled, then the player wins $ \$1$. If the number chosen appears on the bottom of both of the dice, then the player wins $ \$2$. If the number chosen does not appear on the bottom of either of the dice, the player loses $ \$1$. What is the expected return to the player, in dollars, for one roll of the dice? $ \textbf{(A)}\ \minus{}\frac{1}{8}\qquad \textbf{(B)}\ \minus{}\frac{1}{16}\qquad \textbf{(C)}\ 0\qquad \textbf{(D)}\ \frac{1}{16}\qquad \textbf{(E)}\ \frac{1}{8}$

$16$ white and $4$ red balls that are not identical are distributed randomly into $4$ boxes which contain at most $5$ balls. What is the probability that each box contains exactly $1$ red ball? $ \textbf{(A)}\ \dfrac{5}{64} \qquad\textbf{(B)}\ \dfrac{1}{8} \qquad\textbf{(C)}\ \dfrac{4^4}{\binom{16}{4}} \qquad\textbf{(D)}\ \dfrac{5^4}{\binom{20}{4}} \qquad\textbf{(E)}\ \dfrac{3}{32} $

Given three identical $n$- faced dice whose corresponding faces are identically numbered with arbitrary integers. Prove that if they are tossed at random, the probability that the sum of the bottom three face numbers is divisible by three is greater than or equal to $\frac{1}{4}$.

Spinners A and B are spun. On each spinner, the arrow is equally likely to land on each number. What is the probability that the product of the two spinners' numbers is even? [asy] defaultpen(linewidth(1)); draw(unitcircle); draw((1,0)--(-1,0)); draw((0,1)--(0,-1)); draw(shift(3,0)*unitcircle); draw(shift(3,0)*(origin--dir(90))); draw(shift(3,0)*(origin--dir(210))); draw(shift(3,0)*(origin--dir(330))); draw(0.7*dir(200)--0.7*dir(20), linewidth(0.7), EndArrow(7)); draw(shift(3,0)*(0.7*dir(180+65)--0.7*dir(65)), linewidth(0.7), EndArrow(7)); label("$1$", (-0.45,0.1), N); label("$4$", (-0.45,-0.1), S); label("$3$", (0.45,-0.1), S); label("$2$", (0.45,0.1), N); label("$1$", shift(3,0)*(-0.25,0.1), NW); label("$2$", shift(3,0)*(0.25,0.1), NE); label("$3$", shift(3,0)*(0,-0.3), S); label("$A$", (0,-1), S); label("$B$", (3,-1), S); [/asy] $ \textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{1}{3}\qquad\textbf{(C)}\ \frac{1}{2}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{3}{4} $

The center of a square of side length $ 1$ is placed uniformly at random inside a circle of radius $ 1$. Given that we are allowed to rotate the square about its center, what is the probability that the entire square is contained within the circle for some orientation of the square?

In an urn there are one ball marked $1$, two balls marked $2$, and so on, up to $n$ balls marked $n$. Two balls are randomly drawn without replacement. Find the probability that the two balls are assigned the same number.

Kevin has $255$ cookies, each labeled with a unique nonempty subset of $\{1,2,3,4,5,6,7,8\}$. Each day, he chooses one cookie uniformly at random out of the cookies not yet eaten. Then, he eats that cookie, and all remaining cookies that are labeled with a subset of that cookie (for example, if he chooses the cookie labeled with $\{1,2\}$, he eats that cookie as well as the cookies with $\{1\}$ and $\{2\}$). The expected value of the number of days that Kevin eats a cookie before all cookies are gone can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [i]Proposed by Ray Li[/i]

One fair die has faces $ 1$, $ 1$, $ 2$, $ 2$, $ 3$, $ 3$ and another has faces $ 4$, $ 4$, $ 5$, $ 5$, $ 6$, $ 6$. The dice are rolled and the numbers on the top faces are added. What is the probability that the sum will be odd? $ \textbf{(A)}\ \frac{1}{3}\qquad \textbf{(B)}\ \frac{4}{9}\qquad \textbf{(C)}\ \frac{1}{2}\qquad \textbf{(D)}\ \frac{5}{9}\qquad \textbf{(E)}\ \frac{2}{3}$

A positive integer $ N$ with three digits in its base ten representation is chosen at random, with each three digit number having an equal chance of being chosen. The probability that $ \log_2 N$ is an integer is $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 3/899 \qquad \textbf{(C)}\ 1/225 \qquad \textbf{(D)}\ 1/300 \qquad \textbf{(E)}\ 1/450$

Let $\zeta_1, \zeta_2,\ldots$ be identically distributed, independent real-valued random variables with expected value $0$. Suppose that the $\Lambda (\lambda) := \log \mathbb E \exp (\lambda \zeta_i)$ logarithmic moment-generating function always exists for $\lambda\in\mathbb R$ ($\mathbb E$ is the expected value). Furthermore, let $G\colon\mathbb R \rightarrow \mathbb R$ be a function such that $G(x)\leq \min (|x|, x^2)$. Prove that for small $\gamma >0$ the following sequence is bounded: $$\left\{ \mathbb E \exp \left( \gamma l G \left( \frac 1l (\zeta_1+\ldots + \zeta_l)\right)\right)\right\}^{\infty}_{l=1}$$ (translated by j___d)