Let $T=\text{TNFTPP}$. Fermi and Feynman play the game $\textit{Probabicloneme}$ in which Fermi wins with probability $a/b$, where $a$ and $b$ are relatively prime positive integers such that $a/b<1/2$. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play $\textit{Probabicloneme}$ so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is $(T-332)/(2T-601)$. Find the value of $a$.

A $3\times3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is the rotated $90^\circ$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability that the grid is now entirely black? $ \textbf{(A)}\ \dfrac{49}{512} \qquad\textbf{(B)}\ \dfrac{7}{64} \qquad\textbf{(C)}\ \dfrac{121}{1024} \qquad\textbf{(D)}\ \dfrac{81}{512} \qquad\textbf{(E)}\ \dfrac{9}{32} $

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

$6$ people each have a hat. If they shuffle their hats and redistribute them, what is the probability that exactly one person gets their own hat back?

Let $n$ be a integer and $n \ge 3$, and $T_1T_2...T_n$ is a regular n-gon. Distinct $3$ points $T_i , T_j , T_k$ are chosen randomly. Determine the probability of triangle $T_iT_jT_k$ being an acute triangle.

The diagram shows a large circular dart board with four smaller shaded circles each internally tangent to the larger circle. Two of the internal circles have half the radius of the large circle, and are, therefore, tangent to each other. The other two smaller circles are tangent to these circles. If a dart is thrown so that it sticks to a point randomly chosen on the dart board, then the probability that the dart sticks to a point in the shaded area is $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [asy] size(150); defaultpen(linewidth(0.8)); filldraw(circle((0,0.5),.5),gray); filldraw(circle((0,-0.5),.5),gray); filldraw(circle((2/3,0),1/3),gray); filldraw(circle((-2/3,0),1/3),gray); draw(unitcircle); [/asy]

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]

Let $X$ and $Y$ be independent random variables with "Saint-Petersburg" distribution, i.e. for any $k=1,2,\ldots$ their value is $2^k$ with probability $\frac{1}{2^k}$. Show that $X$ and $Y$ can be realized on a sufficiently big probability space such that there exists another pair of independent "Saint-Petersburg" random variables $(X', Y')$ on this space with the property that $X+Y=2X'+Y'I(Y'\le X')$ almost surely (here $I(A)$ denotes the indicator function of the event $A$). (translated by L. Erdős)

A drunken person walks randomly on a tree. Each time, he chooses uniformly at random a neighbouring node and walks there. Show that wherever his starting point and goal are, the expected number of steps the person takes to reach the goal is always an integer. [i]houkai[/i]

Tom, Dick, and Harry are playing a game. Starting at the same time, each of them flips a fair coin repeatedly until he gets his first head, at which point he stops. What is the probability that all three flip their coins the same number of times? $\textbf{(A)}\ \frac{1}{8} \qquad \textbf{(B)}\ \frac{1}{7} \qquad \textbf{(C)}\ \frac{1}{6} \qquad \textbf{(D)}\ \frac{1}{4} \qquad \textbf{(E)}\ \frac{1}{3}$

Shanille O'Keal shoots free throws on a basketball court. She hits the first and misses the second, and thereafter the probability that she hits the next shot is equal to the proportion of shots she has hit so far. What is the probability she hits exactly $50$ of her first $100$ shots?

Real numbers $x,y$, and $z$ are chosen independently and at random from the interval $[0,n]$ for some positive integer $n$. The probability that no two of $x,y$, and $z$ are within $1$ unit of each other is greater than $\tfrac{1}{2}$. What is the smallest possible value of $n$? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11 $

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?

In a new school $40$ percent of the students are freshmen, $30$ percent are sophomores, $20$ percent are juniors, and $10$ percent are seniors. All freshmen are required to take Latin, and $80$ percent of the sophomores, $50$ percent of the juniors, and $20$ percent of the seniors elect to take Latin. The probability that a randomly chosen Latin student is a sophomore is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

In the Duma there are 1600 delegates, who have formed 16000 committees of 80 persons each. Prove that one can find two committees having no fewer than four common members. [i]A. Skopenkov[/i]

[b]10.[/b] Consider a point performing a random walk on a planar triangular lattice and suppose that it moves away with equal probability from any lattice point along any one of the six lattice lines issuing from it. Prove that if the walk is continued indefinitely, then the point will return to its starting point with probability 1. [b](P. 5)[/b]

Alice knows that $3$ red cards and $3$ black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

There are $2$ runners on the perimeter of a regular hexagon, initially located at adjacent vertices. Every second, each of the runners independently moves either one vertex to the left, with probability $\frac{1}{2}$, or one vertex to the right, also with probability $\frac{1}{2}$. Find the probability that after a $2013$ second run (in which runners switch vertices $2013$ times each), the runners end up at adjacent vertices once again.

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} $

$ R_k(m,n)$ is the least number such that for each coloring of $ k$-subsets of $ \{1,2,\dots,R_k(m,n)\}$ with blue and red colors, there is a subset with $ m$ elements such that all of its k-subsets are red or there is a subset with $ n$ elements such that all of its $ k$-subsets are blue. a) If we give a direction randomly to all edges of a graph $ K_n$ then what is the probability that the resultant graph does not have directed triangles? b) Prove that there exists a $ c$ such that $ R_3(4,n)\geq2^{cn}$.

A small apartment building has four doors, with door numbers $1, 2, 3, 4.$ John has $2^4-1=15$ keys, label with of possible nonempty subsets of $\{1,2,3,4\}$, but he forgot which key is which. If an element on the key matches the door number, the key can open the door (e.g. key $\{1,2,4\}$ can open Door 4). He picks a key at random and tries to open Door 1, which fails, so he discards it. John then randomly picks one of his remaining 14 keys and tries to open Door 2, but it doesn't open, so he throws away that key as well. He then randomly selects one of the remaining 13 keys, and tests it on Door 3. What is the probability that it will open? [i]Proposed by dantx5[/i]

Suppose $x$ is a random real number between $1$ and $4$, and $y$ is a random real number between $1$ and $9$. If the expected value of \[ \left\lceil \log_2 x \right\rceil - \left\lfloor \log_3 y \right\rfloor \] can be expressed as $\frac mn$ where $m$ and $n$ are relatively prime positive integers, compute $100m + n$. [i]Proposed by Lewis Chen[/i]

A supermarket has $128$ crates of apples. Each crate contains at least $120$ apples and at most $144$ apples. What is the largest integer $n$ such that there must be at least $n$ crates containing the same number of apples? $\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }24\qquad \textbf{(E) }25$

The faces of a cubical die are marked with the numbers $ 1$, $ 2$, $ 2$, $ 3$, $ 3$, and $ 4$. The faces of a second cubical die are marked with the numbers $ 1$, $ 3$, $ 4$, $ 5$, $ 6$, and $ 8$. Both dice are thrown. What is the probability that the sum of the two top numbers will be $ 5$, $ 7$, or $ 9$ ? $ \textbf{(A)}\ \frac {5}{18} \qquad \textbf{(B)}\ \frac {7}{18} \qquad \textbf{(C)}\ \frac {11}{18} \qquad \textbf{(D)}\ \frac {3}{4} \qquad \textbf{(E)}\ \frac {8}{9}$

Kailey starts with the number $0$, and she has a fair coin with sides labeled $1$ and $2$. She repeatedly flips the coin, and adds the result to her number. She stops when her number is a positive perfect square. What is the expected value of Kailey’s number when she stops? If E is your estimate and A is the correct answer, you will receive $\left\lfloor 25e^{-\frac{5|E-A|}{2} }\right\rfloor$ points.