A rock is dropped off a cliff of height $ h $ As it falls, a camera takes several photographs, at random intervals. At each picture, I measure the distance the rock has fallen. Let the average (expected value) of all of these distances be $ kh $. If the number of photographs taken is huge, find $ k $. That is: what is the time-average of the distance traveled divided by $ h $, dividing by $h$? $ \textbf {(A) } \dfrac{1}{4} \qquad \textbf {(B) } \dfrac{1}{3} \qquad \textbf {(C) } \dfrac{1}{\sqrt{2}} \qquad \textbf {(D) } \dfrac{1}{2} \qquad \textbf {(E) } \dfrac{1}{\sqrt{3}} $ [i]Problem proposed by Ahaan Rungta[/i]

Four distinct points, $A$, $B$, $C$, and $D$, are to be selected from $1996$ points evenly spaced around a circle. All quadruples are equally likely to be chosen. What is the probability that the chord $AB$ intersects the chord $CD$? $\text{(A)}\ \frac 14 \qquad \text{(B)}\ \frac 13 \qquad \text{(C)}\ \frac 12 \qquad \text{(D)}\ \frac 23\qquad \text{(E)}\ \frac 34$

A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\sqrt{2}$ in the center of the square, as shown in the diagram. A circular coin with diameter $1$ is dropped onto the square and lands in a random location where the coin is completely contained within the square, The probability that the coin will cover part of the black region of the square can be written as $\frac{1}{196}(a+b\sqrt{2}+\pi)$, where $a$ and $b$ are positive integers. What is $a+b$? [asy] //Diagram by Samrocksnature draw((0,0)--(8,0)--(8,8)--(0,8)--(0,0)); fill((2,0)--(0,2)--(0,0)--cycle, black); fill((6,0)--(8,0)--(8,2)--cycle, black); fill((8,6)--(8,8)--(6,8)--cycle, black); fill((0,6)--(2,8)--(0,8)--cycle, black); fill((4,6)--(2,4)--(4,2)--(6,4)--cycle, black); filldraw(circle((2.6,3.31),0.47),gray); [/asy] $\textbf{(A) }64 \qquad \textbf{(B) }66 \qquad \textbf{(C) }68 \qquad \textbf{(D) }70 \qquad \textbf{(E) }72$

Each of $2023$ balls is placed in on of $3$ bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls? $\textbf{(A) } \frac{2}{3} \qquad \textbf{(B) } \frac{3}{10} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } \frac{1}{3} \qquad \textbf{(E) } \frac{1}{4}$

For a given number $ n $, let us denote by $ p_n $ the probability that when randomly selecting a pair of integers $ k, m $ satisfying the conditions $ 0 \leq k \leq m \leq 2^n $ (the selection of each pair is equally probable) the number $\binom{m}{k}$ will be even. Calculate $ \lim_{n\to \infty} p_n $.

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?

Contessa is taking a random lattice walk in the plane, starting at $(1,1)$. (In a random lattice walk, one moves up, down, left, or right $1$ unit with equal probability at each step.) If she lands on a point of the form $(6m,6n)$ for $m,n \in \mathbb{Z}$, she ascends to heaven, but if she lands on a point of the form $(6m+3,6n+3)$ for $m,n \in \mathbb{Z}$, she descends to hell. What is the probability she ascends to heaven?

Arnold is studying the prevalence of three health risk factors, denoted by A, B, and C. within a population of men. For each of the three factors, the probability that a randomly selected man in the population as only this risk factor (and none of the others) is 0.1. For any two of the three factors, the probability that a randomly selected man has exactly two of these two risk factors (but not the third) is 0.14. The probability that a randomly selected man has all three risk factors, given that he has A and B is $\tfrac{1}{3}$. The probability that a man has none of the three risk factors given that he does not have risk factor A is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Sophie has $20$ indistinguishable pairs of socks in a laundry bag. She pulls them out one at a time. After pulling out $30$ socks, the expected number of unmatched socks among the socks that she has pulled out can be expressed in simplest form as $\tfrac{m}{n}$. Find $m+n$.

A cube is formed from $n^3$ ($n \ge 2$) unit cubes, each painted white on five randomly selected sides. This cube is dipped into paint remover and broken into the original unit cubes. What is the expected number of these unit cubes with exactly four sides painted white?

Dave rolls a fair six-sided die until a six appears for the first time. Independently, Linda rolls a fair six-sided die until a six appears for the first time. Let $ m$ and $ n$ be relatively prime positive integers such that $ \frac{m}{n}$ is the probability that the number of times Dave rolls his die is equal to or within one of the number of times Linda rolls her die. Find $ m\plus{}n$.

The New York Times Mini Crossword is a $5\times5$ grid with the top left and bottom right corners shaded. Each row and column has a clue given (so that there are $10$ clues total). Jeffrey has a $\frac{1}{2}$ chance of knowing the answer to each clue. What is the probability that he can fill in every unshaded square in the crossword? [asy] size(4cm); for (int i = 0; i < 6; ++i) {draw((i,0)--(i,5)); draw((0,i)--(5,i));} fill((0,4)--(1,4)--(1,5)--(0,5)--cycle, black); fill((5,0)--(5,1)--(4,1)--(4,0)--cycle, black); [/asy] [i]2018 CCA Math Bonanza Individual Round #8[/i]

A box contains 36 pink, 18 blue, 9 green, 6 red, and 3 purple cubes that are identical in size. If a cube is selected at random, what is the probability that it is green? $\textbf{(A)}\ \dfrac{1}{9} \qquad \textbf{(B)}\ \dfrac{1}{8} \qquad \textbf{(C)}\ \dfrac{1}{5} \qquad \textbf{(D)}\ \dfrac{1}{4} \qquad \textbf{(E)}\ \dfrac{9}{70}$

A tree $T$ is given. Starting with the complete graph on $n$ vertices, subgraphs isomorphic to $T$ are erased at random until no such subgraph remains. For what trees does there exist a positive constant $c$ such that the expected number of edges remaining is at least $cn^2$ for all positive integers $n$? [i]David Yang.[/i]

A coin is flipped until there is a head followed by two tails. What is the probability that this will take exactly $12$ flips?

You have to organize a fair procedure to randomly select someone from $ n$ people so that every one of them would be chosen with the probability $ \frac{1}{n}$. You are allowed to choose two real numbers $ 0<p_1<1$ and $ 0<p_2<1$ and order two coins which satisfy the following requirement: the probability of tossing "heads" on the first coin $ p_1$ and the probability of tossing "heads" on the second coin is $ p_2$. Before starting the procedure, you are supposed to announce an upper bound on the total number of times that the two coins are going to be flipped altogether. Describe a procedure that achieves this goal under the given conditions.

The number $21!=51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd? $\textbf{(A)} \frac{1}{21} \qquad \textbf{(B)} \frac{1}{19} \qquad \textbf{(C)} \frac{1}{18} \qquad \textbf{(D)} \frac{1}{2} \qquad \textbf{(E)} \frac{11}{21}$

Let $S$ be the set of integers which are both a multiple of $70$ and a factor of $630{,}000$. A random element $c$ of $S$ is selected. If the probability that there exists an integer $d$ with $\gcd (c,d) = 70$ and $\operatorname{lcm} (c,d) = 630{,}000$ is $\frac mn$ for some relatively prime integers $m$ and $n$, compute $100m+n$. [i]Proposed by Eugene Chen[/i]

Let $p_n(k)$ be the number of permutations of the set $\{1,2,3,\ldots,n\}$ which have exactly $k$ fixed points. Prove that $\sum_{k=0}^nk p_n(k)=n!$.[i](IMO Problem 1)[/i] [b][i]Original formulation [/i][/b] Let $S$ be a set of $n$ elements. We denote the number of all permutations of $S$ that have exactly $k$ fixed points by $p_n(k).$ Prove: (a) $\sum_{k=0}^{n} kp_n(k)=n! \ ;$ (b) $\sum_{k=0}^{n} (k-1)^2 p_n(k) =n! $ [i]Proposed by Germany, FR[/i]

Ali and Veli goes to hunting. The probability that each will successfully hit a duck is $1/2$ on any given shot. During the hunt, Ali shoots $12$ times, and Veli shoots $13$ times. What is the probability that Veli hits more ducks than Ali? $ \textbf{(A)}\ \dfrac 12 \qquad\textbf{(B)}\ \dfrac{13}{25} \qquad\textbf{(C)}\ \dfrac{13}{24} \qquad\textbf{(D)}\ \dfrac{7}{13} \qquad\textbf{(E)}\ \dfrac{3}{4} $

Let $S$ be the set of 81 points $(x, y)$ such that $x$ and $y$ are integers from $-4$ through $4$. Let $A$, $B$, and $C$ be random points chosen independently from $S$, with each of the 81 points being equally likely. (The points $A$, $B$, and $C$ do not have to be different.) Let $K$ be the area of the (possibly degenerate) triangle $ABC$. What is the expected value (average value) of $K^2$ ?

We choose random a unitary polynomial of degree $n$ and coefficients in the set $1,2,...,n!$. Prove that the probability for this polynomial to be special is between $0.71$ and $0.75$, where a polynomial $g$ is called special if for every $k>1$ in the sequence $f(1), f(2), f(3),...$ there are infinitely many numbers relatively prime with $k$.

A non-zero digit is chosen in such a way that the probability of choosing digit $ d$ is $ \log_{10}(d\plus{}1) \minus{} \log_{10} d$. The probability that the digit $ 2$ is chosen is exactly $ \frac12$ the probability that the digit chosen is in the set $ \textbf{(A)}\ \{2,3\} \qquad \textbf{(B)}\ \{3,4\} \qquad \textbf{(C)}\ \{4,5,6,7,8\} \qquad \textbf{(D)}\ \{5,6,7,8,9\} \qquad \textbf{(E)}\ \{4,5,6,7,8,9\}$

Positive integers $ a,b,$ and $ c$ are randomly and independently selected with replacement from the set $ \{ 1,2,3,\dots,2010 \}.$ What is the probability that $ abc \plus{} ab \plus{} a$ is divisible by $ 3$? $ \textbf{(A)}\ \dfrac{1}{3} \qquad\textbf{(B)}\ \dfrac{29}{81} \qquad\textbf{(C)}\ \dfrac{31}{81} \qquad\textbf{(D)}\ \dfrac{11}{27} \qquad\textbf{(E)}\ \dfrac{13}{27}$

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?