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

Let the sequence $a_i$ be defined as $a_{i+1} = 2^{a_i}$. Find the number of integers $1 \le n \le 1000$ such that if $a_0 = n$, then $100$ divides $a_{1000} - a_1$.

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.

Given a point $p$ and a line segment $l$, let $d(p,l)$ be the distance between them. Let $A$, $B$, and $C$ be points in the plane such that $AB=6$, $BC=8$, $AC=10$. What is the area of the region in the $(x,y)$-plane formed by the ordered pairs $(x,y)$ such that there exists a point $P$ inside triangle $ABC$ with $d(P,AB)+x=d(P,BC)+y=d(P,AC)?$

Find the rightmost non-zero digit of the expansion of $(20)(13!)$.

The sequence of integers $\{a_i\}_{i = 0}^{\infty}$ satisfies $a_0 = 3$, $a_1 = 4$, and \[a_{n+2} = a_{n+1} a_n + \left\lceil \sqrt{a_{n+1}^2 - 1} \sqrt{a_n^2 - 1}\right\rceil\] for $n \ge 0$. Evaluate the sum \[\sum_{n = 0}^{\infty} \left(\frac{a_{n+3}}{a_{n+2}} - \frac{a_{n+2}}{a_n} + \frac{a_{n+1}}{a_{n+3}} - \frac{a_n}{a_{n+1}}\right).\]

Let $ABCDEF$ be a regular hexagon, and let $P$ be a point inside quadrilateral $ABCD$. If the area of triangle $PBC$ is $20$, and the area of triangle $PAD$ is $23$, compute the area of hexagon $ABCDEF$.

You have some white one-by-one tiles and some black and white two-bye-one tiles as shown below. There are four different color patterns that can be generated when using these tiles to cover a three-by-one rectangoe by laying these tiles side by side (WWW, BWW, WBW, WWB). How many different color patterns can be generated when using these tiles to cover a ten-by-one rectangle? [asy] import graph; size(5cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((12,0)--(12,1)--(11,1)--(11,0)--cycle); fill((13.49,0)--(13.49,1)--(12.49,1)--(12.49,0)--cycle, black); draw((13.49,0)--(13.49,1)--(14.49,1)--(14.49,0)--cycle); draw((15,0)--(15,1)--(16,1)--(16,0)--cycle); fill((17,0)--(17,1)--(16,1)--(16,0)--cycle, black); [/asy]

Let $S = \{1, 2, \ldots, 2016\}$, and let $f$ be a randomly chosen bijection from $S$ to itself. Let $n$ be the smallest positive integer such that $f^{(n)}(1) = 1$, where $f^{(i)}(x) = f(f^{(i-1)}(x))$. What is the expected value of $n$?

Let $ABCD$ be a parallelogram. Points $X$ and $Y$ lie on segments $AB$ and $AD$ respectively, and $AC$ intersects $XY$ at point $Z$. Prove that \[\frac{AB}{AX} + \frac{AD}{AY} = \frac{AC}{AZ}.\]

Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins, or else indenitely. If Nathaniel goes first, determine the probability that he ends up winning.

Let $\left\{ a_n \right\}$ and $\left\{ b_n \right\}$ be sequences defined recursively by $a_0 =2$; $b_0 = 2$, and $a_{n+1} = a_n \sqrt{1+a_n^2+b_n^2}-b_n$; $b_{n+1} = b_n\sqrt{1+a_n^2+b_n^2} + a_n$. Find the ternary (base 3) representation of $a_4$ and $b_4$.

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.)

Let $ABC$ be an acute triangle with circumcenter $O$, orthocenter $H$, and circumcircle $\Omega$. Let $M$ be the midpoint of $AH$ and $N$ the midpoint of $BH$. Assume the points $M$, $N$, $O$, $H$ are distinct and lie on a circle $\omega$. Prove that the circles $\omega$ and $\Omega$ are internally tangent to each other. [i]Dhroova Aiylam and Evan Chen[/i]

Six unit disks $C_1$, $C_2$, $C_3$, $C_4$, $C_5$, $C_6$ are in the plane such that they don't intersect each other and $C_i$ is tangent to $C_{i+1}$ for $1 \le i \le 6$ (where $C_7 = C_1$). Let $C$ be the smallest circle that contains all six disks. Let $r$ be the smallest possible radius of $C$, and $R$ the largest possible radius. Find $R - r$.

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.

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?

Let $R$ be the region in the Cartesian plane of points $(x,y)$ satisfying $x\geq 0$, $y\geq 0$, and $x+y+\lfloor x\rfloor+\lfloor y\rfloor\leq 5$. Determine the area of $R$.

Let $n$ be a fixed positive integer, and choose $n$ positive integers $a_1, \ldots , a_n$. Given a permutation $\pi$ on the first $n$ positive integers, let $S_{\pi}=\{i\mid \frac{a_i}{\pi(i)} \text{ is an integer}\}$. Let $N$ denote the number of distinct sets $S_{\pi}$ as $\pi$ ranges over all such permutations. Determine, in terms of $n$, the maximum value of $N$ over all possible values of $a_1, \ldots , a_n$. [i]Proposed by James Lin.[/i]

How many real triples $(a,b,c)$ are there such that the polynomial $p(x)=x^4+ax^3+bx^2+ax+c$ has exactly three distinct roots, which are equal to $\tan y$, $\tan 2y$, and $\tan 3y$ for some real number $y$?

7. What is the minimum value of the product $$\prod_{i=1}^6\frac{a_i-a_{i+1}}{a_{i+2}-a_{i+3}}$$ given that $(a_1, a_2, a_3, a_4, a_5, a_6)$ is a permutation of $(1, 2, 3, 4, 5, 6)$? (note $a_7 = a_1, a_8 = a_2 \ldots$) 8. Danielle picks a positive integer $1 \le n \le 2016$ uniformly at random. What is the probability that $\text{gcd}(n, 2015) = 1$? 9. How many $3$-element subsets of the set $\{1, 2, 3, . . . , 19\}$ have sum of elements divisible by $4$?

Convex hexagon $ABCDEF$ is drawn in the plane such that $ACDF$ and $ABDE$ are parallelograms with area 168. $AC$ and $BD$ intersect at $G$. Given that the area of $AGB$ is 10 more than the area of $CGB$, find the smallest possible area of hexagon $ABCDEF$.

4. A square can be divided into four congruent figures as shown: [asy] size(2cm); draw((0,0)--(2,0)--(2,2)--(0,2)--cycle); draw((1,0)--(1,2)); draw((0,1)--(2,1)); [/asy] For how many $n$ with $1 \le n \le 100$ can a unit square be divided into $n$ congruent figures? 5. If $x + 2y - 3z = 7$ and $2x - y + 2z = 6$, determine $8x + y$. 6. Let $ABCD$ be a rectangle, and let $E$ and $F$ be points on segment $AB$ such that $AE = EF = FB$. If $CE$ intersects the line $AD$ at $P$, and $PF$ intersects $BC$ at $Q$, determine the ratio of $BQ$ to $CQ$.

Let $L$ be the length of the altitude to the hypotenuse of a right triangle with legs $5$ and $12$. Find the least integer greater than $L$.

Let $x$ and $y$ be real numbers with $x>y$ such that $x^2y^2+x^2+y^2+2xy=40$ and $xy+x+y=8$. Find the value of $x$.