Suppose that there are initially eight townspeople and one goon. One of the eight townspeople is named Jester. If Jester is sent to jail during some morning, then the game ends immediately in his sole victory. (However, the Jester does not win if he is sent to jail during some night.)
Find the probability that only the Jester wins.
Official shared rules: Start in the morning; $t,g$ count townspeople/goons. Morning: uniformly jail one survivor; stop if $g=0$ or $t=0$. Otherwise evening: uniformly jail one townsperson; stop if $g\ge t$.
Official preamble, page 1: https://hmmt-archive.s3.amazonaws.com/tournaments/2014/nov/thm/problems.pdf
For any positive integers $a$ and $b$, define $a \oplus b$ to be the result when adding $a$ to $b$ in binary (base 2), neglecting any carry-overs. For example, $20 \oplus 14=10100_{2} \oplus 1110_{2}=11010_{2}=26$. (The operation $\oplus$ is called the exclusive or.) Compute the sum
$$
\sum_{k=0}^{2^{2014}-1}\left(k \oplus\left\lfloor\frac{k}{2}\right\rfloor\right)
$$
Here $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.
In rectangle $ABCD$ with area $1$, point $M$ is selected on $\overline{AB}$ and points $X, Y$ are selected on $\overline{CD}$ such that $AX \lt AY$. Suppose that $AM = BM$. Given that the area of triangle $MXY$ is $\frac{1}{2014}$, compute the area of trapezoid $AXYB$.
Supplemental illustration; no additional hypothesis is imposed.
There are $n$ girls $G_{1}, \ldots, G_{n}$ and $n$ boys $B_{1}, \ldots, B_{n}$. A pair $(G_{i}, B_{j})$ is called suitable if and only if girl $G_{i}$ is willing to marry boy $B_{j}$. Given that there is exactly one way to pair each girl with a distinct boy that she is willing to marry, what is the maximal possible number of suitable pairs?
For integers $m, n \geq 1$, let $A(n, m)$ be the number of sequences $(a_{1}, \cdots, a_{n m})$ of integers satisfying the following two properties:
(a) Each integer $k$ with $1 \leq k \leq n$ occurs exactly $m$ times in the sequence $(a_{1}, \cdots, a_{n m})$.
(b) If $i, j$, and $k$ are integers such that $1 \leq i \leq n m$ and $1 \leq j \leq k \leq n$, then $j$ occurs in the sequence $(a_{1}, \cdots, a_{i})$ at least as many times as $k$ does.
For example, if $n=2$ and $m=5$, a possible sequence is $(a_{1}, \cdots, a_{10})=(1,1,2,1,2,2,1,2,1,2)$. On the other hand, the sequence $(a_{1}, \cdots, a_{10})=(1,2,1,2,2,1,1,1,2,2)$ does not satisfy property (2) for $i=5, j=1$, and $k=2$.
Victoria wants to order at least $550$ donuts from Dunkin' Donuts for the HMMT 2014 November contest. However, donuts only come in multiples of twelve. Assuming every twelve donuts cost $\text{USD } 7.49$, what is the minimum amount Victoria needs to pay, in dollars? (Because HMMT is affiliated with MIT, the purchase is tax exempt. Moreover, because of the size of the order, there is no delivery fee.)
Sammy has a wooden board, shaped as a rectangle with length $2^{2014}$ and height $3^{2014}$. The board is divided into a grid of unit squares. A termite starts at either the left or bottom edge of the rectangle, and walks along the gridlines by moving either to the right or upwards, until it reaches an edge opposite the one from which the termite started. Depicted below are two possible paths of the termite.
The termite's path dissects the board into two parts. Sammy is surprised to find that he can still arrange the pieces to form a new rectangle not congruent to the original rectangle. This rectangle has perimeter $P$. How many possible values of $P$ are there?
Let $O_{1}$ and $O_{2}$ be concentric circles with radii $4$ and $6$, respectively. A chord $AB$ is drawn in $O_{1}$ with length $2$. Extend $AB$ to intersect $O_{2}$ in points $C$ and $D$. Find $CD$.
Supplemental illustration; no additional hypothesis is imposed.
A finite sequence of integers $a_1, a_2, \dots, a_n$ is called *regular* if there exists a real number $x$ satisfying
$$
\lfloor kx \rfloor = a_k \quad \text{for } 1 \le k \le n.
$$
Given a regular sequence $a_1, a_2, \dots, a_n$, for $1 \le k \le n$ we say the term $a_k$ is *forced* if the following condition is satisfied: the sequence
$$
a_1, a_2, \dots, a_{k-1}, b
$$
is regular if and only if $b = a_k$. Find the maximum possible number of forced terms in a regular sequence with 1000 terms.
Consider a regular $n$-gon with $n\gt 3$, and call a line acceptable if it passes through the interior of this $n$-gon. Draw $m$ different acceptable lines, so that the $n$-gon is divided into several smaller polygons.
a. Prove that there exists an $m$, depending only on $n$, such that any collection of $m$ acceptable lines results in one of the smaller polygons having 3 or 4 sides.
b. Find the smallest possible $m$ which guarantees that at least one of the smaller polygons will have 3 or 4 sides.
Supplemental illustration; no additional hypothesis is imposed.
Find the number of strictly increasing sequences of nonnegative integers with the following properties:
- The first term is $0$ and the last term is $12$. In particular, the sequence has at least two terms.
- Among any two consecutive terms, exactly one of them is even.
Suppose that $(a_{1}, \ldots, a_{20})$ and $(b_{1}, \ldots, b_{20})$ are two sequences of integers such that the sequence $(a_{1}, \ldots, a_{20}, b_{1}, \ldots, b_{20})$ contains each of the numbers $1, \ldots, 40$ exactly once. What is the maximum possible value of the sum
How many ways can the eight vertices of a three-dimensional cube be colored red and blue such that no two points connected by an edge are both red? Rotations and reflections of a given coloring are considered distinct.
Supplemental illustration; no additional hypothesis is imposed.
Let $ABCD$ be a trapezoid with $AB \parallel CD$ and $\angle D = 90^{\circ}$. Suppose that there is a point $E$ on $CD$ such that $AE = BE$ and that triangles $AED$ and $CEB$ are similar, but not congruent. Given that $\frac{CD}{AB} = 2014$, find $\frac{BC}{AD}$.
Supplemental illustration; no additional hypothesis is imposed.
Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Let $\Gamma$ be the circumcircle of $ABC$, let $O$ be its circumcenter, and let $M$ be the midpoint of minor arc $\widehat{BC}$. Circle $\omega_1$ is internally tangent to $\Gamma$ at $A$, and circle $\omega_2$, centered at $M$, is externally tangent to $\omega_1$ at a point $T$. Ray $AT$ meets segment $BC$ at point $S$, such that $BS - CS = 4/15$. Find the radius of $\omega_2$.
Supplemental illustration; no additional hypothesis is imposed.
