For real numbers $ a$ and $ b$, define $ a\$b\equal{}(a\minus{}b)^2$. What is $ (x\minus{}y)^2\$(y\minus{}x)^2$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ x^2\plus{}y^2 \qquad \textbf{(C)}\ 2x^2 \qquad \textbf{(D)}\ 2y^2 \qquad \textbf{(E)}\ 4xy$

Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints: [list] [*]each cell contains a distinct divisor; [*]the sums of all rows are equal; and [*]the sums of all columns are equal. [/list]

Random sequences $a_1, a_2, . . .$ and $b_1, b_2, . . .$ are chosen so that every element in each sequence is chosen independently and uniformly from the set $\{0, 1, 2, 3, . . . , 100\}$. Compute the expected value of the smallest nonnegative integer $s$ such that there exist positive integers $m$ and $n$ with $$s =\sum^m_{i=1} a_i =\sum^n_{j=1}b_j .$$

Seventeen people correspond by mail with one another-each one with all the rest. In their letters only three different topics are discussed. each pair of correspondents deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic.

Two eight-sided dice each have faces numbered $ 1$ through $ 8$. When the dice are rolled, each face has an equal probability of appearing on the top. What is the probability that the product of the two top numbers is greater than their sum? $ \textbf{(A)}\ \frac{1}{2}\qquad \textbf{(B)}\ \frac{47}{64}\qquad \textbf{(C)}\ \frac{3}{4}\qquad \textbf{(D)}\ \frac{55}{64}\qquad \textbf{(E)}\ \frac{7}{8}$

What is the degree measure of the acute angle formed by lines with slopes $2$ and $\tfrac{1}{3}$? $\textbf{(A)}~30\qquad\textbf{(B)}~37.5\qquad\textbf{(C)}~45\qquad\textbf{(D)}~52.5\qquad\textbf{(E)}~60$

What is the greatest value of $\sin x \cos y + \sin y \cos z + \sin z \cos x$, where $x,y,z$ are real numbers? $ \textbf{(A)}\ \sqrt 2 \qquad\textbf{(B)}\ \dfrac 32 \qquad\textbf{(C)}\ \dfrac {\sqrt 3}2 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 3 $

Let $f(x)$ be a polynomial of nonzero degree. Can it happen that for any real number $a$, an even number of real numbers satisfy the equation $f(x) = a$?

Each field of the $n \times n$ array has been colored either red or blue, with the following conditions met: $\bullet$ if a row and a column contain the same number of red fields, the field at their intersection is red; $\bullet$ if a row and a column contain different numbers of red cells, the field at their intersection is blue. Prove that the total number of blue cells is even.

Show that the number of 5-tuples ($a$, $b$, $c$, $d$, $e$) such that $abcde = 5(bcde + acde + abde + abce + abcd)$ is odd

The center of the circumcircle of the acute triangle $ABC$ is $O$. The line $AO$ intersects $BC$ at $D$. On the sides $AB$ and $AC$ of the triangle, choose points $E$ and $F$, respectively, so that the points $A, E, D, F$ lie on the same circle. Let $E'$ and $F'$ projections of points $E$ and $F$ on side $BC$ respectively. Prove that length of the segment $E'F'$ does not depend on the position of points $E$ and $F$.

Find the number of nonnegative integers $n < 29$ such that there exists positive integers $x,y$ where $$x^2+5xy-y^2$$ has remainder $n$ when divided by $29$.

[u]Set 4[/u] [b]p10.[/b] Eve has nine letter tiles: three $C$’s, three $M$’s, and three $W$’s. If she arranges them in a random order, what is the probability that the string “$CMWMC$” appears somewhere in the arrangement? [b]p11.[/b] Bethany’s Batteries sells two kinds of batteries: $C$ batteries for $\$4$ per package, and $D$ batteries for $\$7$ per package. After a busy day, Bethany looks at her ledger and sees that every customer that day spent exactly $\$2021$, and no two of them purchased the same quantities of both types of battery. Bethany also notes that if any other customer had come, at least one of these two conditions would’ve had to fail. How many packages of batteries did Bethany sell? [b]p12.[/b] A deck of cards consists of $30$ cards labeled with the integers $1$ to $30$, inclusive. The cards numbered $1$ through $15$ are purple, and the cards numbered $16$ through $30$ are green. Lilith has an expansion pack to the deck that contains six indistinguishable copies of a green card labeled with the number $32$. Lilith wants to pick from the expanded deck a hand of two cards such that at least one card is green. Find the number of distinguishable hands Lilith can make with this deck. PS. You should use hide for answers.

Find all integer solutions to the equation $y^k = x^2 + x$, where $k$ is a natural number greater than $1$.

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Given that $a + 19b = 3$ and $a + 1019b = 5$, what is $a + 2019b$? [b]p2.[/b] How many multiples of $3$ are there between $2019$ and $2119$, inclusive? [b]p3.[/b] What is the maximum number of quadrilaterals a $12$-sided regular polygon can be quadrangulated into? Here quadrangulate means to cut the polygon along lines from vertex to vertex, none of which intersect inside the polygon, to form pieces which all have exactly $4$ sides. [b]p4.[/b] What is the value of $|2\pi - 7| + |2\pi - 6|$, rounded to the nearest hundredth? [b]p5.[/b] In the town of EMCCxeter, there is a $30\%$ chance that it will snow on Saturday, and independently, a $40\%$ chance that it will snow on Sunday. What is the probability that it snows exactly once that weekend, as a percentage? [b]p6.[/b] Define $n!$ to be the product of all integers between $1$ and $n$ inclusive. Compute $\frac{2019!}{2017!} \times \frac{2016!}{2018!}$ . [b]p7.[/b] There are $2019$ people standing in a row, and they are given positions $1$, $2$, $3$, $...$, $2019$ from left to right. Next, everyone in an odd position simultaneously leaves the row, and the remaining people are assigned new positions from $1$ to $1009$, again from left to right. This process is then repeated until one person remains. What was this person's original position? [b]p8.[/b] The product $1234\times 4321$ contains exactly one digit not in the set $\{1, 2, 3, 4\}$. What is this digit? [b]p9.[/b] A quadrilateral with positive area has four integer side lengths, with shortest side $1$ and longest side $9$. How many possible perimeters can this quadrilateral have? [b]p10.[/b] Define $s(n)$ to be the sum of the digits of $n$ when expressed in base $10$, and let $\gamma (n)$ be the sum of $s(d)$ over all natural number divisors $d$ of $n$. For instance, $n = 11$ has two divisors, $1$ and $11$, so $\gamma (11) = s(1) + s(11) = 1 + (1 + 1) = 3$. Find the value of $\gamma (2019)$. [b]p11.[/b] How many five-digit positive integers are divisible by $9$ and have $3$ as the tens digit? [b]p12.[/b] Adam owns a large rectangular block of cheese, that has a square base of side length $15$ inches, and a height of $4$ inches. He wants to remove a cylindrical cheese chunk of height $4$, by making a circular hole that goes through the top and bottom faces, but he wants the surface area of the leftover cheese block to be the same as before. What should the diameter of his hole be, in inches? [i]Αddendum on 1/26/19: the hole must have non-zero diameter. [/i] [b]p13.[/b] Find the smallest prime that does not divide $20! + 19! + 2019!$. [b]p14.[/b] Convex pentagon $ABCDE$ has angles $\angle ABC = \angle BCD = \angle DEA = \angle EAB$ and angle $\angle CDE = 60^o$. Given that $BC = 3$, $CD = 4$, and $DE = 5$, find $EA$. [i]Addendum on 1/26/19: ABCDE is specified to be convex. [/i] [b]p15.[/b] Sophia has $3$ pairs of red socks, $4$ pairs of blue socks, and $5$ pairs of green socks. She picks out two individual socks at random: what is the probability she gets a pair with matching color? [b]p16.[/b] How many real roots does the function $f(x) = 2019^x - 2019x - 2019$ have? [b]p17.[/b] A $30-60-90$ triangle is placed on a coordinate plane with its short leg of length $6$ along the $x$-axis, and its long leg along the $y$-axis. It is then rotated $90$ degrees counterclockwise, so that the short leg now lies along the $y$-axis and long leg along the $x$-axis. What is the total area swept out by the triangle during this rotation? [b]p18.[/b] Find the number of ways to color the unit cells of a $2\times 4$ grid in four colors such that all four colors are used and every cell shares an edge with another cell of the same color. [b]p19.[/b] Triangle $\vartriangle ABC$ has centroid $G$, and $X, Y,Z$ are the centroids of triangles $\vartriangle BCG$, $\vartriangle ACG$, and $\vartriangle ABG$, respectively. Furthermore, for some points $D,E, F$, vertices $A,B,C$ are themselves the centroids of triangles $\vartriangle DBC$, $\vartriangle ECA$, and $\vartriangle FAB$, respectively. If the area of $\vartriangle XY Z = 7$, what is the area of $\vartriangle DEF$? [b]p20.[/b] Fhomas orders three $2$-gallon jugs of milk from EMCCBay for his breakfast omelette. However, every jug is actually shipped with a random amount of milk (not necessarily an integer), uniformly distributed between $0$ and $2$ gallons. If Fhomas needs $2$ gallons of milk for his breakfast omelette, what is the probability he will receive enough milk? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For a positive integer $n$, we say an $n$-[i]shuffling[/i] is a bijection $\sigma: \{1,2, \dots , n\} \rightarrow \{1,2, \dots , n\}$ such that there exist exactly two elements $i$ of $\{1,2, \dots , n\}$ such that $\sigma(i) \neq i$. Fix some three pairwise distinct $n$-shufflings $\sigma_1,\sigma_2,\sigma_3$. Let $q$ be any prime, and let $\mathbb{F}_q$ be the integers modulo $q$. Consider all functions $f:(\mathbb{F}_q^n)^n\to\mathbb{F}_q$ that satisfy, for all integers $i$ with $1 \leq i \leq n$ and all $x_1,\ldots x_{i-1},x_{i+1}, \dots ,x_n, y, z\in\mathbb{F}_q^n$, \[f(x_1, \ldots ,x_{i-1}, y, x_{i+1}, \ldots , x_n) +f(x_1, \ldots ,x_{i-1}, z, x_{i+1}, \ldots , x_n) = f(x_1, \ldots ,x_{i-1}, y+z, x_{i+1}, \ldots , x_n), \] and that satisfy, for all $x_1,\ldots,x_n\in\mathbb{F}_q^n$ and all $\sigma\in\{\sigma_1,\sigma_2,\sigma_3\}$, \[f(x_1,\ldots,x_n)=-f(x_{\sigma(1)},\ldots,x_{\sigma(n)}).\] For a given tuple $(x_1,\ldots,x_n)\in(\mathbb{F}_q^n)^n$, let $g(x_1,\ldots,x_n)$ be the number of different values of $f(x_1,\ldots,x_n)$ over all possible functions $f$ satisfying the above conditions. Pick $(x_1,\ldots,x_n)\in(\mathbb{F}_q^n)^n$ uniformly at random, and let $\varepsilon(q,\sigma_1,\sigma_2,\sigma_3)$ be the expected value of $g(x_1,\ldots,x_n)$. Finally, let \[\kappa(\sigma_1,\sigma_2,\sigma_3)=-\lim_{q \to \infty}\log_q\left(-\ln\left(\frac{\varepsilon(q,\sigma_1,\sigma_2,\sigma_3)-1}{q-1}\right)\right).\] Pick three pairwise distinct $n$-shufflings $\sigma_1,\sigma_2,\sigma_3$ uniformly at random from the set of all $n$-shufflings. Let $\pi(n)$ denote the expected value of $\kappa(\sigma_1,\sigma_2,\sigma_3)$. Suppose that $p(x)$ and $q(x)$ are polynomials with real coefficients such that $q(-3) \neq 0$ and such that $\pi(n)=\frac{p(n)}{q(n)}$ for infinitely many positive integers $n$. Compute $\frac{p\left(-3\right)}{q\left(-3\right)}$.