Let $ M \equal{} \{ x_1..., x_{30}\}$ a set which consists of 30 distinct positive numbers, let $ A_n,$ $ 1 \leq n \leq 30,$ the sum of all possible products with $ n$ elements each of the set $ M.$ Prove if $ A_{15} > A_{10},$ then $ A_1 > 1.$

Ten balls numbered $1$ to $10$ are in a jar. Jack reaches into the jar and randomly removes one of the balls. Then Jill reaches into the jar and randomly removes a different ball. The probability that the sum of the two numbers on the balls removed is even is $\text{(A)}\ \frac{4}{9} \qquad \text{(B)}\ \frac{9}{19} \qquad \text{(C)}\ \frac{1}{2} \qquad \text{(D)}\ \frac{10}{19} \qquad \text{(E)}\ \frac{5}{9}$

Jorn wants to cheat at the role play: he intends to cheat the sides to re-label its two octahedra, so that each of the numbers from $1$ to $16$ has the same probability as the sum of the dice occurs. So that the game master does not notice this so easily, he only wants to use numbers from $0$ to $8$ , if necessary several times or not at all. Is this possible?

We are given the finite sets $ X$, $ A_1$, $ A_2$, $ \dots$, $ A_{n \minus{} 1}$ and the functions $ f_i: \ X\rightarrow A_i$. A vector $ (x_1,x_2,\dots,x_n)\in X^n$ is called [i]nice[/i], if $ f_i(x_i) \equal{} f_i(x_{i \plus{} 1})$, for each $ i \equal{} 1,2,\dots,n \minus{} 1$. Prove that the number of nice vectors is at least \[ \frac {|X|^n}{\prod\limits_{i \equal{} 1}^{n \minus{} 1} |A_i|}. \]

A gambling student tosses a fair coin. She gains $1$ point for each head that turns up, and gains $2$ points for each tail that turns up. Prove that the probability of the student scoring [i]exactly[/i] $n$ points is $\frac{1}{3}\cdot\left(2+\left(-\frac{1}{2}\right)^{n}\right)$.

A point $ P$ is chosen at random in the interior of equilateral triangle $ ABC$. What is the probability that $ \triangle ABP$ has a greater area than each of $ \triangle ACP$ and $ \triangle BCP$? $ \textbf{(A)}\ \frac{1}{6} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{1}{3} \qquad \textbf{(D)}\ \frac{1}{2} \qquad \textbf{(E)}\ \frac{2}{3}$

Suppose 2012 reals are selected independently and at random from the unit interval $[0,1]$, and then written in nondecreasing order as $x_1\le x_2\le\cdots\le x_{2012}$. If the probability that $x_{i+1} - x_i \le \frac{1}{2011}$ for $i=1,2,\ldots,2011$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m,n$, find the remainder when $m+n$ is divided by 1000. [i]Victor Wang.[/i]

When 7 fair standard 6-sided dice are thrown, the probability that the sum of the numbers on the top faces is 10 can be written as $$\frac{n}{6^7},$$where $n$ is a positive integer. What is $n$? $\textbf{(A) } 42 \qquad \textbf{(B) } 49 \qquad \textbf{(C) } 56 \qquad \textbf{(D) } 63 \qquad \textbf{(E) } 84 $

Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular icosahedron is written a nonnegative integer such that the sum of all $20$ integers is $39.$ Show that there are two faces that share a vertex and have the same integer written on them.

A particle performing a random walk on the integer points of the semi-axis $x \ge 0$ moves a distance $1$ to the right with probability $a$, and to the left with probability $b$, and stands still in the remaining cases (if $x = 0$, it stands still instead of moving to the left). Determine the steady-state probability distribution, and also the expectation of $x$ and $x^2$ over a long time, if the particle starts at the point $0$.

There are 81 grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point $P$ is the center of the square. Given that point $Q$ is randomly chosen from among the other 80 points, what is the probability that line $PQ$ is a line of symmetry for the square? [asy] size(130); defaultpen(fontsize(11)); int i, j; for(i=0; i<9; i=i+1) { for(j=0; j<9; j=j+1) if((i==4) && (j==4)) { dot((i,j),linewidth(5)); } else { dot((i,j),linewidth(3)); } } dot("$P$",(4,4),NE); draw((0,0)--(0,8)--(8,8)--(8,0)--cycle); [/asy] $\textbf{(A) } \frac{1}{5} \qquad\textbf{(B) } \frac{1}{4} \qquad\textbf{(C) } \frac{2}{5} \qquad\textbf{(D) } \frac{9}{20} \qquad\textbf{(E) } \frac{1}{2}$

For a given natural number $n$, two players randomly (uniformly distributed) select a common number $0 \le j \le n$, and then each of them independently randomly selects a subset of $\{1,2, \cdots, n \}$ with $j$ elements. Let $p_n$ be the probability that the same set was chosen. Prove that \[ \sum_{k=1}^{n} p_k = 2 \log{n} + 2 \gamma - 1 + o(1), \quad (n \to \infty),\] where $\gamma$ is the Euler constant.

A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0),$ $(2020, 0),$ $(2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to the nearest tenth$?$ $\textbf{(A) } 0.3 \qquad \textbf{(B) } 0.4 \qquad \textbf{(C) } 0.5 \qquad \textbf{(D) } 0.6 \qquad \textbf{(E) } 0.7$

Two different numbers are randomly selected from the set ${ - 2, -1, 0, 3, 4, 5}$ and multiplied together. What is the probability that the product is $0$? $\textbf{(A) }\dfrac{1}{6}\qquad\textbf{(B) }\dfrac{1}{5}\qquad\textbf{(C) }\dfrac{1}{4}\qquad\textbf{(D) }\dfrac{1}{3}\qquad \textbf{(E) }\dfrac{1}{2}$

As part of his effort to take over the world, Edward starts producing his own currency. As part of an effort to stop Edward, Alex works in the mint and produces $1$ counterfeit coin for every $99$ real ones. Alex isn't very good at this, so none of the counterfeit coins are the right weight. Since the mint is not perfect, each coin is weighed before leaving. If the coin is not the right weight, then it is sent to a lab for testing. The scale is accurate $95\%$ of the time, $5\%$ of all the coins minted are sent to the lab, and the lab's test is accurate $90\%$ of the time. If the lab says a coin is counterfeit, what is the probability that it really is?

Bob the bomb-defuser has stumbled upon an active bomb. He opens it up, and finds the red and green wires conveniently located for him to cut. Being a seasoned member of the bomb-squad, Bob quickly determines that it is the green wire that he should cut, and puts his wirecutters on the green wire. But just before he starts to cut, the bomb starts to count down, ticking every second. Each time the bomb ticks, starting at time $t=15$ seconds, Bob panics and has a certain chance to move his wirecutters to the other wire. However, he is a rational man even when panicking, and has a $\frac{1}{2t^2}$ chance of switching wires at time t, regardless of which wire he is about to cut. When the bomb ticks at $t=1$, Bob cuts whatever wire his wirecutters are on, without switching wires. What is the probability that Bob cuts the green wire?

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} $

Alex starts at the origin $O$ of a hexagonal lattice. Every second, he moves to one of the six vertices adjacent to the vertex he is currently at. If he ends up at $X$ after $2018$ moves, then let $p$ be the probability that the shortest walk from $O$ to $X$ (where a valid move is from a vertex to an adjacent vertex) has length $2018$. Then $p$ can be expressed as $\tfrac{a^m-b}{c^n}$, where $a$, $b$, and $c$ are positive integers less than $10$; $a$ and $c$ are not perfect squares; and $m$ and $n$ are positive integers less than $10000$. Find $a+b+c+m+n$.

Ten tiles numbered $1$ through $10$ are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square? $\textbf{(A)}\ \frac{1}{10}\qquad \textbf{(B)}\ \frac{1}{6}\qquad \textbf{(C)}\ \frac{11}{60}\qquad \textbf{(D)}\ \frac{1}{5}\qquad \textbf{(E)}\ \frac{7}{30}$

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.

Terence Tao is playing rock-paper-scissors. Because his mental energy is focused on solving the twin primes conjecture, he uses the following very simple strategy:    ·He plays rock first.    ·On each subsequent turn, he plays a different move than the previous one, each with probability ½. What is the probability that his 5th move will be rock?

Let $K$ be a finite field with $q$ elements and let $n \ge q$ be an integer. Find the probability that by choosing an $n$-th degree polynomial with coefficients in $K,$ it doesn't have any root in $K.$

Compute the expected sum of elements in a subset of $\{1, 2, 3, . . . , 2020\}$ (including the empty set) chosen uniformly at random.

$(1)$ Start with $a$ white balls and $b$ black balls. $(2)$ Draw one ball at random. $(3)$ If the ball is white, then stop. Otherwise, add two black balls and go to step $2$. Let $S$ be the number of draws before the process terminates. For the cases $a = b = 1$ and $a = b = 2$ only, find $a_n = P(S = n), b_n = P(S \le n), \lim_{n\to\infty} b_n$, and the expectation value of the number of balls drawn: $E(S) =\displaystyle\sum_{n\ge1} na_n.$

There are $15$ people, including Petruk, Gareng, and Bagong, which will be partitioned into $6$ groups, randomly, that consists of $3, 3, 3, 2, 2$, and $2$ people (orders are ignored). Determine the probability that Petruk, Gareng, and Bagong are in a group.