MathNet 2013

How many orderings $\left(a_{1}, \ldots, a_{8}\right)$ of $(1,2, \ldots, 8)$ exist such that $a_{1}-a_{2}+a_{3}-a_{4}+a_{5}-a_{6}+a_{7}-a_{8}=0$?

Compute the prime factorization of $1007021035035021007001$. (You should write your answer in the form $p_{1}^{e_{1}} p_{2}^{e_{2}} \ldots p_{k}^{e_{k}}$, where $p_{1}, \ldots, p_{k}$ are distinct prime numbers and $e_{1}, \ldots, e_{k}$ are positive integers.)

Let $x$ and $y$ be real numbers with $x \gt y$ such that $x^{2} y^{2} + x^{2} + y^{2} + 2 x y = 40$ and $x y + x + y = 8$. Find the value of $x$.

Plot points $A$, $B$, $C$ at coordinates $(0,0)$, $(0,1)$, and $(1,1)$ in the plane, respectively. Let $S$ denote the union of the two line segments $AB$ and $BC$. Let $X_{1}$ be the area swept out when Bobby rotates $S$ counterclockwise $45$ degrees about point $A$. Let $X_{2}$ be the area swept out when Calvin rotates $S$ clockwise $45$ degrees about point $A$. Find $\frac{X_{1}+X_{2}}{2}$.

Supplemental illustration (AI-generated):

Define the sequence $\{x_{i}\}_{i \geq 0}$ by $x_{0}=x_{1}=x_{2}=1$ and $x_{k}=\frac{x_{k-1}+x_{k-2}+1}{x_{k-3}}$ for $k\gt 2$. Find $x_{2013}$.

Let $N$ be the largest positive integer that can be expressed as a 2013-digit base $-4$ number. What is the remainder when $N$ is divided by $210$?

In a game, there are three indistinguishable boxes; one box contains two red balls, one contains two blue balls, and the last contains one ball of each color. To play, Raj first predicts whether he will draw two balls of the same color or two of different colors. Then, he picks a box, draws a ball at random, looks at the color, and replaces the ball in the same box. Finally, he repeats this; however, the boxes are not shuffled between draws, so he can determine whether he wants to draw again from the same box. Raj wins if he predicts correctly; if he plays optimally, what is the probability that he will win?

Let $N$ be a positive integer whose decimal representation contains $11235$ as a contiguous substring, and let $k$ be a positive integer such that $10^{k} \gt N$. Find the minimum possible value of
$$
\frac{10^{k}-1}{\operatorname{gcd}\left(N, 10^{k}-1\right)}
$$

Consider triangle $ABC$ with side lengths $AB = 4$, $BC = 7$, and $AC = 8$. Let $M$ be the midpoint of segment $AB$, and let $N$ be the point on the interior of segment $AC$ that also lies on the circumcircle of triangle $MBC$. Compute $BN$.

Let $\omega = \cos \frac{2\pi}{727} + i \sin \frac{2\pi}{727}$. The imaginary part of the complex number
$$
\prod_{k=8}^{13}\left(1+\omega^{3^{k-1}}+\omega^{2 \cdot 3^{k-1}}\right)
$$
is equal to $\sin \alpha$ for some angle $\alpha$ between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$, inclusive. Find $\alpha$.

Jarris the triangle is playing in the $(x, y)$ plane. Let his maximum $y$ coordinate be $k$. Given that he has side lengths $6$, $8$, and $10$ and that no part of him is below the $x$-axis, find the minimum possible value of $k$.

Supplemental illustration (AI-generated):

Let $b(n)$ be the number of digits in the base $-4$ representation of $n$. Evaluate $\sum_{i=1}^{2013} b(i)$.

For how many integers $1 \leq k \leq 2013$ does the decimal representation of $k^{k}$ end with a $1$?

Let $A_{1} A_{2} A_{3} A_{4} A_{5} A_{6}$ be a convex hexagon such that $A_{i} A_{i+2} \parallel A_{i+3} A_{i+5}$ for $i=1,2,3$ (we take $A_{i+6}=A_{i}$ for each $i$). Segment $A_{i} A_{i+2}$ intersects segment $A_{i+1} A_{i+3}$ at $B_{i}$, for $1 \leq i \leq 6$, as shown. Furthermore, suppose that $\triangle A_{1} A_{3} A_{5} \cong \triangle A_{4} A_{6} A_{2}$. Given that $[A_{1} B_{5} B_{6}]=1$, $[A_{2} B_{6} B_{1}]=4$, and $[A_{3} B_{1} B_{2}]=9$ (by $[X Y Z]$ we mean the area of $\triangle X Y Z$), determine the area of hexagon $B_{1} B_{2} B_{3} B_{4} B_{5} B_{6}$.

The real numbers $x, y, z$ satisfy $0 \leq x \leq y \leq z \leq 4$. If their squares form an arithmetic progression with common difference $2$, determine the minimum possible value of $|x-y| + |y-z|$.

Two cars are driving directly towards each other such that one is twice as fast as the other. The distance between their starting points is $4$ miles. When the two cars meet, how many miles is the faster car from its starting point?

Let $\omega_{1}$ and $\omega_{2}$ be circles with centers $O_{1}$ and $O_{2}$, respectively, and radii $r_{1}$ and $r_{2}$, respectively. Suppose that $O_{2}$ is on $\omega_{1}$. Let $A$ be one of the intersections of $\omega_{1}$ and $\omega_{2}$, and $B$ be one of the two intersections of line $O_{1} O_{2}$ with $\omega_{2}$. If $A B = O_{1} A$, find all possible values of $\frac{r_{1}}{r_{2}}$.

Evaluate the expression

where the digit $2$ appears $2013$ times.

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?

A standard 52-card deck contains cards of 4 suits and 13 numbers, with exactly one card for each pairing of suit and number. If Maya draws two cards with replacement from this deck, what is the probability that the two cards have the same suit or have the same number, but not both?