Triangle $ABC$ lies entirely in the first quadrant of the Cartesian plane, and its sides have slopes $63$, $73$, $97$. Suppose the curve $\mathcal V$ with equation $y=(x+3)(x^2+3)$ passes through the vertices of $ABC$. Find the sum of the slopes of the three tangents to $\mathcal V$ at each of $A$, $B$, $C$. [i]Proposed by Akshaj[/i]

Let $z_0+z_1+z_2+\cdots$ be an infinite complex geometric series such that $z_0=1$ and $z_{2013}=\dfrac 1{2013^{2013}}$. Find the sum of all possible sums of this series.

The vertices of a regular nonagon are colored such that $1)$ adjacent vertices are different colors and $2)$ if $3$ vertices form an equilateral triangle, they are all different colors. Let $m$ be the minimum number of colors needed for a valid coloring, and n be the total number of colorings using $m$ colors. Determine $mn$. (Assume each vertex is distinguishable.)

In a rectangular box $ABCDEFGH$ with edge lengths $AB = AD = 6$ and $AE = 49$, a plane slices through point $A$ and intersects edges $BF$, $FG$, $GH$, $HD$ at points $P$, $Q$, $R$, $S$ respectively. Given that $AP = AS$ and $PQ = QR = RS$, find the area of pentagon $APQRS$.

What is the largest factor of $130000$ that does not contain the digit $0$ or $5$?

Theseus starts at the point $(0, 0)$ in the plane. If Theseus is standing at the point $(x, y)$ in the plane, he can step one unit to the north to point $(x, y+1)$, one unit to the west to point $(x-1, y)$, one unit to the south to point $(x, y-1)$, or one unit to the east to point $(x+1, y)$. After a sequence of more than two such moves, starting with a step one unit to the south (to point $(0, -1)$), Theseus finds himself back at the point $(0, 0)$. He never visited any point other than $(0, 0)$ more than once, and never visited the point $(0, 0)$ except at the start and end of this sequence of moves. Let $X$ be the number of times that Theseus took a step one unit to the north, and then a step one unit to the west immediately afterward. Let $Y$ be the number of times that Theseus took a step one unit to the west, and then a step one unit to the north immediately afterward. Prove that $|X - Y| = 1$. [i]Mitchell Lee[/i]

In rectangle $ABCD$, points $E$ and $F$ lie on sides $AB$ and $CD$ respectively such that both $AF$ and $CE$ are perpendicular to diagonal $BD$. Given that $BF$ and $DE$ separate $ABCD$ into three polygons with equal area, and that $EF = 1$, find the length of $BD$.

Let $ABCD$ be a quadrilateral inscribed in the unit circle such that $\angle BAD$ is $30$ degrees. Let $m$ denote the minimum value of $CP + PQ + CQ$, where $P$ and $Q$ may be any points lying along rays $AB$ and $AD$, respectively. Determine the maximum value of $m$.

Suppose that there are $16$ variables $\{a_{i,j}\}_{0\leq i,j\leq 3}$, each of which may be $0$ or $1$. For how many settings of the variables $a_{i,j}$ do there exist positive reals $c_{i,j}$ such that the polynomial \[f(x,y)=\sum_{0\leq i,j\leq 3}a_{i,j}c_{i,j}x^iy^j\] $(x,y\in\mathbb{R})$ is bounded below?

34. Find the sum of the ages of everyone who wrote a problem for this year’s HMMT November contest. If your answer is $X$ and the actual value is $Y$ , your score will be $\text{max}(0, 20 - |X - Y|)$ 35. Find the total number of occurrences of the digits $0, 1 \ldots , 9$ in the entire guts round (the official copy). If your answer is $X$ and the actual value is $Y$ , your score will be $\text{max}(0, 20 - \frac{|X-Y|}{2})$ 36. Find the number of positive integers less than $1000000$ which are less than or equal to the sum of their proper divisors. If your answer is $X$ and the actual value is $Y$, your score will be $\text{max}(0, 20 - 80|1 - \frac{X}{Y}|)$ rounded to the nearest integer.

Let \[ A = \frac{1}{6}((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3) \]. Compute $2^A$.

Denote by $\mathbb{N}$ the positive integers. Let $f:\mathbb{N} \rightarrow \mathbb{N}$ be a function such that, for any $w,x,y,z \in \mathbb{N}$, \[ f(f(f(z)))f(wxf(yf(z)))=z^{2}f(xf(y))f(w). \] Show that $f(n!) \ge n!$ for every positive integer $n$. [i]Pakawut Jiradilok[/i]

The side lengths of a triangle are distinct positive integers. One of the side lengths is a multiple of $42,$ and another is a multiple of $72$. What is the minimum possible length of the third side?

Suppose $P (x)$ is a polynomial with real coefficients such that $P (t) = P (1)t^2 + P (P (1))t + P (P (P (1)))$ for all real numbers $t$. Compute the largest possible value of $P(P(P(P(1))))$.

How many ways are there to color the vertices of a triangle red, green, blue, or yellow such that no two vertices have the same color? Rotations and reflections are considered distinct.

Find all positive integers $n$ for which there do not exist $n$ consecutive composite positive integers less than $n!$.

Let $ABC$ be a triangle with $AB = 13, BC = 14, $and$ CA = 15$. Suppose $PQRS$ is a square such that $P$ and $R$ lie on line $BC, Q$ lies on line $CA$, and $S$ lies on line $AB$. Compute the side length of this square.

Each cell of a $3 $ × $3$ grid is labeled with a digit in the set {$1, 2, 3, 4, 5$} Then, the maximum entry in each row and each column is recorded. Compute the number of labelings for which every digit from $1$ to $5$ is recorded at least once.

Find the number of positive integers $j\leq 3^{2013}$ such that \[j=\sum_{k=0}^m\left((-1)^k\cdot 3^{a_k}\right)\] for some strictly increasing sequence of nonnegative integers $\{a_k\}$. For example, we may write $3=3^1$ and $55=3^0-3^3+3^4$, but $4$ cannot be written in this form.

A rectangular pool table has vertices at $(0, 0) (12, 0) (0, 10),$ and $(12, 10)$. There are pockets only in the four corners. A ball is hit from $(0, 0)$ along the line $y = x$ and bounces off several walls before eventually entering a pocket. Find the number of walls that the ball bounces off of before entering a pocket.

The numbers $1, 2\ldots11$ are arranged in a line from left to right in a random order. It is observed that the middle number is larger than exactly one number to its left. Find the probability that it is larger than exactly one number to its right.

Spencer is making burritos, each of which consists of one wrap and one filling. He has enough filling for up to four beef burritos and three chicken burritos. However, he only has five wraps for the burritos; in how many orders can he make exactly five burritos?

([b]7[/b]) Evaluate the limit $ \lim_{n\rightarrow\infty} n^{\minus{}\frac{1}{2}\left(1\plus{}\frac{1}{n}\right)} \left(1^1\cdot2^2\cdot\cdots\cdot n^n\right)^{\frac{1}{n^2}}$.

I have $8$ unit cubes of different colors, which I want to glue together into a $2\times 2\times 2$ cube. How many distinct $2\times 2\times 2$ cubes can I make? Rotations of the same cube are not considered distinct, but reflections are.

Let $z$ be a non-real complex number with $z^{23}=1$. Compute \[\sum_{k=0}^{22}\dfrac{1}{1+z^k+z^{2k}}.\]