Alice the ant starts at vertex $A$ of regular hexagon $ABCDEF$ and moves either right or left each move with equal probability. After $35$ moves, what is the probability that she is on either vertex $A$ or $C$? [i]2015 CCA Math Bonanza Lightning Round #5.3[/i]

There are $20$ balls in a bag. $a$ of them are red, $b$ of them are white, and $c$ of them are black. It is known that $ \bullet$ if we double the white balls, the probability of drawing one red ball is $\dfrac 1{25}$ less than the probability of drawing one red ball at the beginning, and $ \bullet$ if we remove all red balls, the probability of drawing one white ball is $\dfrac 1{16}$ more than the probability of drawing one white ball at the beginning. Find $a,b,c$.

A set of three points is randomly chosen from the grid shown. Each three point set has the same probability of being chosen. What is the probability that the points lie on the same straight line? [asy]unitsize(.5cm); defaultpen(linewidth(.8pt)); dotfactor=3; pair[] dotted={(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2)}; dot(dotted);[/asy]$ \textbf{(A)}\ \frac {1}{21}\qquad \textbf{(B)}\ \frac {1}{14}\qquad \textbf{(C)}\ \frac {2}{21}\qquad \textbf{(D)}\ \frac {1}{7}\qquad \textbf{(E)}\ \frac {2}{7}$

The numbers $1, 2, \dots , n$ are written on a blackboard and then erased via the following process:[list] [*] Before any numbers are erased, a pair of numbers is chosen uniformly at random and circled. [*] Each minute for the next $n -1$ minutes, a pair of numbers still on the blackboard is chosen uniformly at random and the smaller one is erased. [*] In minute $n$, the last number is erased. [/list] What is the probability that the smaller circled number is erased before the larger? [i]Proposed by Ethan Tan[/i]

Let $n$ be a positive integer and let $\mathcal{A}$ be a family of non-empty subsets of $\{1, 2, \ldots, n \}$ such that if $A \in \mathcal{A}$ and $A$ is subset of a set $B\subseteq \{1, 2, \ldots, n\}$, then $B$ is also in $\mathcal{A}$. Show that the function $$f(x):=\sum_{A \in \mathcal{A}} x^{|A|}(1-x)^{n-|A|}$$ is strictly increasing for $x \in (0,1)$.

You are playing a game in which you have $3$ envelopes, each containing a uniformly random amount of money between $0$ and $1000$ dollars. (That is, for any real $0 \leq a < b \leq 1000$, the probability that the amount of money in a given envelope is between $a$ and $b$ is $\frac{b-a}{1000}$.) At any step, you take an envelope and look at its contents. You may choose either to keep the envelope, at which point you finish, or discard it and repeat the process with one less envelope. If you play to optimize your expected winnings, your expected winnings will be $E$. What is $\lfloor E\rfloor,$ the greatest integer less than or equal to $E$? [i]Author: Alex Zhu[/i]

A pair of 8-sided dice have sides numbered 1 through 8. Each side has the same probability (chance) of landing face up. The probability that the product of the two numbers that land face-up exceeds 36 is $\textbf{(A)}\ \dfrac{5}{32} \qquad \textbf{(B)}\ \dfrac{11}{64} \qquad \textbf{(C)}\ \dfrac{3}{16} \qquad \textbf{(D)}\ \dfrac{1}{4} \qquad \textbf{(E)}\ \dfrac{1}{2}$

Horses X, Y and Z are entered in a three-horse race in which ties are not possible. If the odds against X winning are $3-to-1$ and the odds against Y winning are $2-to-3$, what are the odds against Z winning? (By "[i]odds against H winning are p-to-q[/i]" we mean that probability of H winning the race is $\frac{q}{p+q}$.) $ \textbf{(A)}\ 3-to-20\qquad\textbf{(B)}\ 5-to-6\qquad\textbf{(C)}\ 8-to-5\qquad\textbf{(D)}\ 17-to-3\qquad\textbf{(E)}\ 20-to-3 $

Let $n\ge3$ be an integer. A regular $n$-gon $P$ is given. We randomly select three distinct vertices of $P$. The probability that these three vertices form an isosceles triangle is $1/m$, where $m$ is an integer. How many such integers $n\le 2024$ are there? \[\rm a. ~674\qquad \mathrm b. ~675\qquad \mathrm c. ~682 \qquad\mathrm d. ~684\qquad\mathrm e. ~685\]

For integer $n\geq2$, let $x_1, x_2, \ldots, x_n$ be real numbers satisfying \[x_1+x_2+\ldots+x_n=0, \qquad \text{and}\qquad x_1^2+x_2^2+\ldots+x_n^2=1.\]For each subset $A\subseteq\{1, 2, \ldots, n\}$, define\[S_A=\sum_{i\in A}x_i.\](If $A$ is the empty set, then $S_A=0$.) Prove that for any positive number $\lambda$, the number of sets $A$ satisfying $S_A\geq\lambda$ is at most $2^{n-3}/\lambda^2$. For which choices of $x_1, x_2, \ldots, x_n, \lambda$ does equality hold?

For an even positive integer $n$ Kevin has a tape of length $4n$ with marks at $-2n,-2n+1,\ldots,2n-1,2n$. He then randomly picks $n$ points in the set $-n,-n+1,-n+2,\ldots,n-1,n$ and places a stone on each of these points. We call a stone 'stuck' if it is on $2n$ or $-2n$, or either all the points to the right, or all the points to the left, all contain stones. Then, every minute, Kevin shifts the unstruck stones in the following manner: [list] [*]He picks an unstuck stone uniformly at random and then flips a fair coin. [*]If the coin came up heads, he then moves that stone and every stone in the largest contiguous set containing that stone one point to the left. If the coin came up tails, he moves every stone in that set one point right instead. [*]He repeats until all the stones are stuck.[/list] Let $p_n$ be the probability that at the end of the process there are exactly $k$ stones in the right half. Evaluate \[\dfrac{p_{n-1}-p_{n-2}+p_{n-3}+\ldots+p_3-p_2+p_1}{p_{n-1}+p_{n-2}+p_{n-3}+\ldots+p_3+p_2+p_1}\] in terms of $n$.

You roll three fair six-sided dice. Given that the highest number you rolled is a $5$, the expected value of the sum of the three dice can be written as $\tfrac ab$ in simplest form. Find $a+b$.

Six distinct positive integers are randomly chosen between $1$ and $2011;$ inclusive. The probability that some pair of the six chosen integers has a difference that is a multiple of $5 $ is $n$ percent. Find $n.$

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.

Urn A contains 4 white balls and 2 red balls. Urn B contains 3 red balls and 3 black balls. An urn is randomly selected, and then a ball inside of that urn is removed. We then repeat the process of selecting an urn and drawing out a ball, without returning the first ball. What is the probability that the first ball drawn was red, given that the second ball drawn was black?

Tom is searching for the $6$ books he needs in a random pile of $30$ books. What is the expected number of books must he examine before finding all $6$ books he needs?

Let $ S$ be the set of all points $ (x,y)$ in the coordinate plane such that $ 0\le x\le \frac \pi2$ and $ 0\le y\le \frac \pi2$. What is the area of the subset of $ S$ for which \[ \sin^2 x \minus{} \sin x\sin y \plus{} \sin^2 y\le \frac 34? \]$ \textbf{(A) } \frac {\pi^2}9 \qquad \textbf{(B) } \frac {\pi^2}8 \qquad \textbf{(C) } \frac {\pi^2}6\qquad \textbf{(D) } \frac {3\pi^2}{16} \qquad \textbf{(E) } \frac {2\pi^2}9$

Let $S$ S be a point chosen at random from the interior of the square $ABCD$, which has side $AB$ and diagonal $AC$. Let $P$ be the probability that the segments $AS$, $SB$, and $AC$ are congruent to the sides of a triangle. Then $P$ can be written as $\dfrac{a-\pi\sqrt{b}-\sqrt{c}}{d}$ where $a,b,c,$ and $d$ are all positive integers and $d$ is as small as possible. Find $ab+cd$.

A solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, no two of which match; otherwise the drawing continues until the bag is empty. The probability that the bag will be emptied is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$

Real numbers $x,y,$ and $z$ are chosen from the interval $[-1,1]$ independently and uniformly at random. What is the probability that $$\vert{x}\vert+\vert{y}\vert+\vert{z}\vert+\vert{x+y+z}\vert=\vert{x+y}\vert+\vert{y+z}\vert+\vert{z+x}\vert?$$

Let $S$ be the square one of whose diagonals has endpoints $(0.1,0.7)$ and $(-0.1,-0.7)$. A point $v=(x,y)$ is chosen uniformly at random over all pairs of real numbers $x$ and $y$ such that $0\le x \le 2012$ and $0 \le y \le 2012$. Let $T(v)$ be a translated copy of $S$ centered at $v$. What is the probability that the square region determined by $T(v)$ contains exactly two points with integer coordinates in its interior? $ \textbf{(A)}\ 0.125\qquad\textbf{(B)}\ 0.14\qquad\textbf{(C)}\ 0.16\qquad\textbf{(D)}\ 0.25\qquad\textbf{(E)}\ 0.32 $

Ben has a big blackboard, initially empty, and Francisco has a fair coin. Francisco flips the coin $2013$ times. On the $n^{\text{th}}$ flip (where $n=1,2,\dots,2013$), Ben does the following if the coin flips heads: (i) If the blackboard is empty, Ben writes $n$ on the blackboard. (ii) If the blackboard is not empty, let $m$ denote the largest number on the blackboard. If $m^2+2n^2$ is divisible by $3$, Ben erases $m$ from the blackboard; otherwise, he writes the number $n$. No action is taken when the coin flips tails. If probability that the blackboard is empty after all $2013$ flips is $\frac{2u+1}{2^k(2v+1)}$, where $u$, $v$, and $k$ are nonnegative integers, compute $k$. [i]Proposed by Evan Chen[/i]

Let $a,b \in [0,1], c \in [-1,1]$ be reals chosen independently and uniformly at random. What is the probability that $p(x) = ax^2+bx+c$ has a root in $[0,1]$?

Let $H$ be the unit hemisphere $\{(x,y,z):x^2+y^2+z^2=1,z\geq 0\}$, $C$ the unit circle $\{(x,y,0):x^2+y^2=1\}$, and $P$ the regular pentagon inscribed in $C$. Determine the surface area of that portion of $H$ lying over the planar region inside $P$, and write your answer in the form $A \sin\alpha + B \cos\beta$, where $A,B,\alpha,\beta$ are real numbers.

Aws plays a solitaire game on a fifty-two card deck: whenever two cards of the same color are adjacent, he can remove them. Aws wins the game if he removes all the cards. If Aws starts with the cards in a random order, what is the probability for him to win?