Let $n$ be a positive integer. For any $k$, denote by $a_k$ the number of permutations of $\{1,2,\dots,n\}$ with exactly $k$ disjoint cycles. (For example, if $n=3$ then $a_2=3$ since $(1)(23)$, $(2)(31)$, $(3)(12)$ are the only such permutations.) Evaluate \[ a_n n^n + a_{n-1} n^{n-1} + \dots + a_1 n. \][i]Proposed by Sammy Luo[/i]

[i]25 problems for 30 minutes[/i] [b]p1.[/b] Compute $2^2 + 0 \cdot 0 + 2^2 + 3^3$. [b]p2.[/b] How many total letters (not necessarily distinct) are there in the names Jerry, Justin, Jackie, Jason, and Jeffrey? [b]p3.[/b] What is the remainder when $20232023$ is divided by $50$? [b]p4.[/b] Let $ABCD$ be a square. The fraction of the area of $ABCD$ that is the area of the intersection of triangles $ABD$ and $ABC$ can be expressed as $\frac{a}{b}$ , where $a$ and $b$ relatively prime positive integers. Find $a + b$. [b]p5.[/b] Raymond is playing basketball. He makes a total of $15$ shots, all of which are either worth $2$ or $3$ points. Given he scored a total of $40$ points, how many $2$-point shots did he make? [b]p6.[/b] If a fair coin is flipped $4$ times, the probability that it lands on heads more often than tails is $\frac{a}{b}$ , where $a$ and $b$ relatively prime positive integers. Find $a + b$. [b]p7.[/b] What is the sum of the perfect square divisors of $640$? [b]p8.[/b] A regular hexagon and an equilateral triangle have the same perimeter. The ratio of the area between the hexagon and equilateral triangle can be expressed in the form $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]p9.[/b] If a cylinder has volume $1024\pi$, radius of $r$ and height $h$, how many ordered pairs of integers $(r, h)$ are possible? [b]p10.[/b] Pump $A$ can fill up a balloon in $3$ hours, while pump $B$ can fill up a balloon in $5$ hours. Pump $A$ starts filling up a balloon at $12:00$ PM, and pump $B$ is added alongside pump $A$ at a later time. If the balloon is completely filled at $2:00$ PM, how many minutes after $12:00$ PM was Pump $B$ added? [b]p11.[/b] For some positive integer $k$, the product $81 \cdot k$ has $20$ factors. Find the smallest possible value of $k$. [b]p12.[/b] Two people wish to sit in a row of fifty chairs. How many ways can they sit in the chairs if they do not want to sit directly next to each other and they do not want to sit with exactly one empty chair between them? [b]p13.[/b] Let $\vartriangle ABC$ be an equilateral triangle with side length $2$ and $M$ be the midpoint of $BC$. Let $P$ be a point in the same plane such that $2PM = BC$. The minimum value of $AP$ can be expressed as $\sqrt{a}-b$, where $a$ and $b$ are positive integers such that $a$ is not divisible by any perfect square aside from $1$. Find $a + b$. [b]p14.[/b] What are the $2022$nd to $2024$th digits after the decimal point in the decimal expansion of $\frac{1}{27}$ , expressed as a $3$ digit number in that order (i.e the $2022$nd digit is the hundreds digit, $2023$rd digit is the tens digit, and $2024$th digit is the ones digit)? [b]p15.[/b] After combining like terms, how many terms are in the expansion of $(xyz+xy+yz+xz+x+y+z)^{20}$? [b]p16.[/b] Let $ABCD$ be a trapezoid with $AB \parallel CD$ where $AB > CD$, $\angle B = 90^o$, and $BC = 12$. A line $k$ is dropped from $A$, perpendicular to line $CD$, and another line $\ell$ is dropped from $C$, perpendicular to line $AD$. $k$ and $\ell$ intersect at $X$. If $\vartriangle AXC$ is an equilateral triangle, the area of $ABCD$ can be written as $m\sqrt{n}$, where $m$ and $n$ are positive integers such that $n$ is not divisible by any perfect square aside from $1$. Find $m + n$. [b]p17.[/b] If real numbers $x$ and $y$ satisfy $2x^2 + y^2 = 8x$, maximize the expression $x^2 + y^2 + 4x$. [b]p18.[/b] Let $f(x)$ be a monic quadratic polynomial with nonzero real coefficients. Given that the minimum value of $f(x)$ is one of the roots of $f(x)$, and that $f(2022) = 1$, there are two possible values of $f(2023)$. Find the larger of these two values. [b]p19.[/b] I am thinking of a positive integer. After realizing that it is four more than a multiple of $3$, four less than a multiple of $4$, four more than a multiple of 5, and four less than a multiple of $7$, I forgot my number. What is the smallest possible value of my number? [b]p20.[/b] How many ways can Aston, Bryan, Cindy, Daniel, and Evan occupy a row of $14$ chairs such that none of them are sitting next to each other? [b]p21.[/b] Let $x$ be a positive real number. The minimum value of $\frac{1}{x^2} +\sqrt{x}$ can be expressed in the form \frac{a}{b^{(c/d)}} , where $a$, $b$, $c$, $d$ are all positive integers, $a$ and $b$ are relatively prime, $c$ and $d$ are relatively prime, and $b$ is not divisible by any perfect square. Find $a + b + c + d$. [b]p22.[/b] For all $x > 0$, the function $f(x)$ is defined as $\lfloor x \rfloor \cdot (x + \{x\})$. There are $24$ possible $x$ such that $f(x)$ is an integer between $2000$ and $2023$, inclusive. If the sum of these $24$ numbers equals $N$, then find $\lfloor N \rfloor$. Note: Recall that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$, called the floor function. Also, $\{x\}$ is defined as $x - \lfloor x \rfloor$, called the fractional part function. [b]p23.[/b] Let $ABCD$ be a rectangle with $AD = 1$. Let $P$ be a point on diagonal $\overline{AC}$, and let $\omega$ and $\xi$ be the circumcircles of $\vartriangle APB$ and $\vartriangle CPD$, respectively. Line $\overleftrightarrow{AD}$ is extended, intersecting $\omega$ at $X$, and $\xi$ at $Y$ . If $AX = 5$ and $DY = 2$, find $[ABCD]^2$. Note: $[ABCD]$ denotes the area of the polygon $ABCD$. [b]p24.[/b] Alice writes all of the three-digit numbers on a blackboard (it’s a pretty big blackboard). Let $X_a$ be the set of three-digit numbers containing a somewhere in its representation, where a is a string of digits. (For example, $X_{12}$ would include $12$, $121$, $312$, etc.) If Bob then picks a value of $a$ at random so $0 \le a \le 999$, the expected number of elements in $X_a$ can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find$ m + n$. [b]p25.[/b] Let $f(x) = x^5 + 2x^4 - 2x^3 + 4x^2 + 5x + 6$ and $g(x) = x^4 - x^3 + x^2 - x + 1$. If $a$, $b$, $c$, $d$ are the roots of $g(x)$, then find $f(a) + f(b) + f(c) + f(d)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $n$ rectangles drawn on the rectangular sheet of paper with the sides of the rectangles parallel to the sheet sides. The rectangles do not have pairwise common interior points. Prove that after cutting out the rectangles the sheet will split into not more than $n+1$ part.

Dima has 100 rocks with pairwise distinct weights. He also has a strange pan scales: one should put exactly 10 rocks on each side. Call a pair of rocks {\it clear} if Dima can find out which of these two rocks is heavier. Find the least possible number of clear pairs.

Consider the real numbers $a_1,a_2,...,a_{2n}$ whose sum is equal to $0$. Prove that among pairs $(a_i,a_j) , i<j$ where $ i,j \in \{1,2,...,2n\} $ .there are at least $2n-1$ pairs with the property that $a_i+a_j\ge 0$.

On a one-way street, an unending sequence of cars of width $a$, length $b$ passes with velocity $v$. The cars are separated by the distance $c$. A pedestrian crosses the street perpendicularly with velocity $w$, without paying attention to the cars. [b](a)[/b] What is the probability that the pedestrian crosses the street uninjured? [b](b)[/b] Can he improve this probability by crossing the road in a direction other than perpendicular?

One moving point in the coordinate plane can move right or up one position. $N$ is a number of all paths : paths that moving point starts from $(0, 0)$, without passing $(1, 0), (2, 1), . . . , (n, n-1)$ and moves $2n$ times to $(n, n)$. $a_k$ is a number of special paths : paths include in $N$, but $k$th moves to the right, $k+1$th moves to the up. find $$\frac{1}{N} (a_1+a_2+ . . . + a_{2n-1})$$

Let $a_k$ be the number of nonnegative integers $ n $ with the properties: a) $n\in[0, 10^k)$ has exactly $ k $ digits, such that he zeroes on the first positions of $ n $ are included in the decimal writting. b) the digits of $ n $ can be permutated such that the new number is divisible by $11.$ Show that $a_{2m}=10a_{2m-1}$ for every $m\in\mathbb{N}.$

[u]Chessboards [/u] Joe B. is playing with some chess pieces on a $6\times 6$ chessboard. Help him find out some things. [b]p1.[/b] Joe B. first places the black king in one corner of the board. In how many of the $35$ remaining squares can he place a white bishop so that it does not check the black king? [b]p2.[/b] Joe B. then places a white king in the opposite corner of the board. How many total ways can he place one black bishop and one white bishop so that neither checks the king of the opposite color? [b]p3.[/b] Joe B. now clears the board. How many ways can he place $3$ white rooks and $3$ black rooks on the board so that no two rooks of opposite color can attack each other? [b]p4.[/b] Joe B. is frustrated with chess. He breaks the board, leaving a $4\times 4$ board, and throws $3$ black knights and $3$ white kings at the board. Miraculously, they all land in distinct squares! What is the expected number of checks in the resulting position? (Note that a knight can administer multiple checks and a king can be checked by multiple knights.) [b]p5.[/b] Suppose that at some point Joe B. has placed $2$ black knights on the original board, but gets bored of chess. He now decides to cover the $34$ remaining squares with $17$ dominos so that no two overlap and the dominos cover the entire rest of the board. For how many initial arrangements of the two pieces is this possible? Note: Chess is a game played with pieces of two colors, black and white, that players can move between squares on a rectangular grid. Some of the pieces move in the following ways: $\bullet$ Bishop: This piece can move any number of squares diagonally if there are no other pieces along its path. $\bullet$ Rook: This piece can move any number of squares either vertically or horizontally if there are no other pieces along its path. $\bullet$ Knight: This piece can move either two squares along a row and one square along a column or two squares along a column and one square along a row. $\bullet$ King: This piece can move to any open adjacent square (including diagonally). If a piece can move to a square occupied by a king of the opposite color, we say that it is checking the king. If a piece moves to a square occupied by another piece, this is called attacking.

Two concentric circles are divided by $n$ radii into $2n$ parts. Two parts are called neighbors (of each other) if they share either a common side or a common arc. Initially, there are $4n + 1$ frogs inside the parts. At each second, if there are three or more frogs inside one part, then three of the frogs in the part will jump to its neighbors, with one to each neighbor. Prove that in a finite amount of time, for any part either there are frogs in the part or there are frogs in each of its neighbors

In chess, a king threatens another king if, and only if, they are on neighboring squares, whether horizontally, vertically, or diagonally . Find the greatest amount of kings that can be placed on a $12 \times 12$ board such that each king threatens just another king. Here, we are not considering part colors, that is, consider that the king are all, say, white, and that kings of the same color can threaten each other.

Each rational point on a real line is assigned an integer. Prove that there is a segment such that the sum of the numbers at its endpoints does not exceed twice the number at its midpoint.

Let $A_1 A_2...A_n$ be a convex $n$-gon. What’s the number of $m$-gons with vertices from $A_1,A_2,...,A_n$ such that between each two adjacent vertices of the $m$-gon there are at least $k$ vertices from the $n$-gon?

A block of size $ a\times b\times c$ is composed of $ 1\times 1\times 2$ domino blocks. Assuming that each of the three possible directions of domino blocks occurs equally many times, what are the possible values of $ a$, $ b$, $ c$?

$110$ teams participate in a volleyball tournament. Every team has played every other team exactly once (there are no ties in volleyball). Turns out that in any set of $55$ teams, there is one which has lost to no more than $4$ of the remaining $54$ teams. Prove that in the entire tournament, there is a team that has lost to no more than $4$ of the remaining $109$ teams.

On a chessboard, Po controls a white queen and plays, in alternate turns, against an invisible black king (there are only those two pieces on the board). The king cannot move to a square where he would be in check, neither capture the queen. Every time the king makes a move, Po receives a message from beyond that tells which direction the king has moved (up, right, up-right, etc). His goal is to make the king unable to make a movement. Can Po reach his goal with at most $150$ moves, regardless the starting position of the pieces?

Karlsson-on-the-Roof has $1000$ jars of jam. The jars are not necessarily identical; each contains no more than $\dfrac{1}{100}$-th of the total amount of the jam. Every morning, Karlsson chooses any $100$ jars and eats the same amount of the jam from each of them. Prove that Karlsson can eat all the jam. [i](8 points)[/i]

[u]Fermi Questions[/u] [b]p1.[/b] What is $\sin (1000)$? (note: that's $1000$ radians, not degrees) [b]p2.[/b] In liters, what is the volume of $10$ million US dollars' worth of gold? [b]p3.[/b] How many trees are there on Earth? [b]p4.[/b] How many prime numbers are there between $10^8$ and $10^9$? [b]p5.[/b] What is the total amount of time spent by humans in spaceflight? [b]p6.[/b] What is the global domestic product (total monetary value of all goods and services produced in a country's borders in a year) of Bangladesh in US dollars? [b]p7.[/b] How much time does the average American spend eating during their lifetime, in hours? [b]p8.[/b] How many CHMMC-related emails did the directors receive or send in the last month? [u]Suspiciously Familiar. . .[/u] [b]p9.[/b] Suppose a farmer learns that he will die at the end of the year (day $365$, where today is day $0$) and that he has $100$ sheep. He decides to sell all his sheep on one day, and that his utility is given by $ab$ where $a$ is the money he makes by selling the sheep (which always have a fixed price) and $b$ is the number of days he has left to enjoy the profit; i.e., $365 - k$ where $k$ is the day number. If every day his sheep breed and multiply their numbers by $(421 + b)/421$ (yes, there are small, fractional sheep), on which day should he sell out? [b]p10.[/b] Suppose in your sock drawer of $14$ socks there are $5$ different colors and $3$ different lengths present. One day, you decide you want to wear two socks that have either different colors or different lengths but not both. Given only this information, what is the maximum number of choices you might have? [u]I'm So Meta Even This Acronym[/u] [b]p11.[/b] Let $\frac{s}{t}$ be the answer of problem $13$, written in lowest terms. Let $\frac{p}{q}$ be the answer of problem $12$, written in lowest terms. If player $1$ wins in problem $11$, let $n = q$. Otherwise, let $n = p$. Two players play a game on a connected graph with $n$ vertices and $t$ edges. On each player's turn, they remove one edge of the graph, and lose if this causes the graph to become disconnected. Which player (first or second) wins? [b]p12.[/b] Let $\frac{s}{t}$ be the answer of problem $13$, written in lowest terms. If player $1$ wins in problem $11$, let $n = t$. Otherwise, let $n = s$. Find the maximum value of $$\frac{x^n}{1 + \frac12 x + \frac14 x^2 + ...+ \frac{1}{2^{2n}} x^{2n}}$$ for $x > 0$. [b]p13.[/b] Let $\frac{p}{q}$ be the answer of problem $12$, written in lowest terms. Let $y$ be the largest integer such that $2^y$ divides $p$. If player $1$ wins in problem $11$, let $z = q$. Otherwise, let $z = p$. Suppose that $a_1 = 1$ and $$a_{n+1} = a_n -\frac{z}{n + 2}+\frac{2z}{n + 1}-\frac{z}{n}$$ What is $a_y$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Ali and Naqi are playing a game. At first, they have Polynomial $P(x) = 1+x^{1398}$. Naqi starts. In each turn one can choice natural number $k \in [0,1398]$ in his trun, and add $x^k$ to the polynomial. For example after 2 moves $P$ can be : $P(x) = x^{1398} + x^{300} + x^{100} +1$. If after Ali's turn, there exist $t \in R$ such that $P(t)<0$ then Ali loses the game. Prove that Ali can play forever somehow he never loses the game!

Let $n$ be a positive integer. In a $(2n+1)\times (2n+1)$ board, each grid is dyed white or black. In each row and each column, if the number of white grids is smaller than the number of black grids, then we mark all white grids. If the number of white grids is bigger than the number of black grids, then we mark all black grids. Let $a$ be the number of black grids, and $b$ be the number of white grids, $c$ is the number of marked grids. In this example of $3\times 3$ table, $a=3$, $b=6$, $c=4$. (forget about my watermark) Proof that no matter how is the dyeing situation in the beginning, there is always $c\geq\frac{1}{2}\min\{a,b\}$.

A $2^{2014} + 1$ by $2^{2014} + 1$ grid has some black squares filled. The filled black squares form one or more snakes on the plane, each of whose heads splits at some points but never comes back together. In other words, for every positive integer $n$ greater than $2$, there do not exist pairwise distinct black squares $s_1$, $s_2$, \dots, $s_n$ such that $s_i$ and $s_{i+1}$ share an edge for $i=1,2, \dots, n$ (here $s_{n+1}=s_1$). What is the maximum possible number of filled black squares? [i]Proposed by David Yang[/i]

Consider a matrix of size $n\times n$ whose entries are real numbers of absolute value not exceeding $1$. The sum of all entries of the matrix is $0$. Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding $C$ in absolute value. [i]Proposed by Marcin Kuczma, Poland[/i]

You are given an odd number $n\ge 3$. For every pair of integers $(i, j)$ with $1\le i \le j \le n$ there is a domino, with $i$ written on one its end and with $j$ written on another (there are $\frac{n(n+1)}{2}$ domino overall). Amin took this dominos and started to put them in a row so that numbers on the adjacent sides of the dominos are equal. He has put $k$ dominos in this way, got bored and went away. After this Anton came to see this $k$ dominos, and he realized that he can't put all the remaining dominos in this row by the rules. For which smallest value of $k$ is this possible? [i]Oleksii Masalitin[/i]

Lines parallel to the sides of an equilateral triangle are drawn so that they cut each of the sides into $10$ equal segments and the triangle into $100$ congruent triangles. Each of these $100$ triangles is called a “cell”. Also lines parallel to each of the sides of the original triangle are drawn through each of the vertices of the original triangle. The cells between any two adjacent parallel lines form a “stripe”. What is the maximum number of cells that can be chosen so that no two chosen cells belong to one stripe? (R Zhenodarov)

Consider a random string $s$ of $10^{2020}$ base-ten digits (there can be leading zeroes). We say a substring $s' $ (which has no leading zeroes) is self-locating if $s' $ appears in $s$ at index $s' $ where the string is indexed at $ 1$. For example the substring $11$ in the string “$122352242411$” is selflocating since the $11$th digit is $ 1$ and the $12$th digit is $ 1$. Let the expected number of self-locating substrings in s be $G$. Compute $\lfloor G \rfloor$.