[b]p1.[/b] Find all prime factors of $8051$. [b]p2.[/b] Simplify $$[\log_{xyz}(x^z)][1 + \log_x y + \log_x z],$$ where $x = 628$, $y = 233$, $z = 340$. [b]p3.[/b] In prokaryotes, translation of mRNA messages into proteins is most often initiated at start codons on the mRNA having the sequence AUG. Assume that the mRNA is single-stranded and consists of a sequence of bases, each described by a single letter A,C,U, or G. Consider the set of all pieces of bacterial mRNA six bases in length. How many such mRNA sequences have either no A’s or no U’s? [b]p4.[/b] What is the smallest positive $n$ so that $17^n + n$ is divisible by $29$? [b]p5.[/b] The legs of the right triangle shown below have length $a = 255$ and $b = 32$. Find the area of the smaller rectangle (the one labeled $R$). [img]https://cdn.artofproblemsolving.com/attachments/c/d/566f2ce631187684622dfb43f36c7e759e2f34.png[/img] [b]p6.[/b] A $3$ dimensional cube contains ”cubes” of smaller dimensions, ie: faces ($2$-cubes),edges ($1$-cubes), and vertices ($0$-cubes). How many 3-cubes are in a $5$-cube? PS. You had better use hide for answers.

Mario has a deck of seven pairs of matching number cards and two pairs of matching Jokers, for a total of $18$ cards. He shuffles the deck, then draws the cards from the top one by one until he holds a pair of matching Jokers. The expected number of complete pairs that Mario holds at the end (including the Jokers) is $\frac{m}{n}$ , where $m$, $n$ are positive integers and gcd $(m,n) = 1$. Find $100m + n$.

Suppose $V= \mathbb{Z}_2^n$ and for a vector $x=(x_1,..x_n)$ in $V$ and permutation $\sigma$.We have $x_{\sigma}=(x_{\sigma(1)},...,x_{\sigma(n)})$ Suppose $ n=4k+2,4k+3$ and $f:V \to V$ is injective and if $x$ and $y$ differ in more than $n/2$ places then $f(x)$ and $f(y)$ differ in more than $n/2$ places. Prove there exist permutaion $\sigma$ and vector $v$ that $f(x)=x_{\sigma}+v$

Divide each side of a triangle into $50$ equal parts, and each point of the division is joined to the opposite vertex by a segment. Calculate the number of intersection points determined by these segments. Clarification : the vertices of the original triangle are not considered points of intersection or division.

We will say that a plane is [i]well-colored[/i] in several colors if it is divided into convex polygons with an area of at least $1/1000$ and each polygon is colored in one color. Points lying on the border of several polygons can be colored in any of their colors. Are there convex is a $9$-gon $R$ and a good coloring of the plane in $7$ colors such that in any polygon obtained from $R$ by a translate to any vector, all colors occupy the same area ($1/7$ of the area of $R$)?

For a natural number $n$ denote by Map $( n )$ the set of all functions $f : \{ 1 , 2 , 3 , \cdots , n \} \rightarrow \{ 1 , 2 , 3 , \cdots , n \} . $ For $ f , g \in $ Map$( n ) , f \circ g $ denotes the function in Map $( n )$ that sends $x \rightarrow f ( g ( x ) ) . $ \\ \\ $(a)$ Let $ f \in$ Map $( n ) . $ If for all $x \in \{ 1 , 2 , 3 , \cdots , n \} f ( x ) \neq x , $ show that $ f \circ f \neq f $ \\$(b)$ Count the number of functions $ f \in$ Map $( n )$ such that $ f \circ f = f $

Let ${n}$ and $k$ be positive integers. There are given ${n}$ circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwise distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of $n$ distinct colors so that each color is used at least once and exactly $k$ distinct colors occur on each circle. Find all values of $n\geq 2$ and $k$ for which such a coloring is possible. [i]Proposed by Horst Sewerin, Germany[/i]

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]

Consider a convex $100$-gon $A_1A_2...A_{100}$. Draw the diagonal $A_{43}A_{81}$ which divides it into two convex polygons $P_1,P_2$. How many vertices and how diagonals, has each of the polygons $P_1,P_2$?

Let the operation $ f$ of $ k$ variables defined on the set $ \{ 1,2,\ldots,n \}$ be called $ \textit{friendly}$ toward the binary relation $ \rho$ defined on the same set if \[ f(a_1,a_2,\ldots,a_k) \;\rho\ \;f(b_1,b_2,\ldots,b_k)\] implies $ a_i \; \rho \ b_i$ for at least one $ i,1\leq i \leq k$. Show that if the operation $ f$ is friendly toward the relations "equal to" and "less than," then it is friendly toward all binary relations. [i]B. Csakany[/i]

[i]25 problems for 30 minutes[/i] [b]p1.[/b] Alisha has $6$ cupcakes and Tyrone has $10$ brownies. Tyrone gives some of his brownies to Alisha so that she has three times as many desserts as Tyrone. How many desserts did Tyrone give to Alisha? [b]p2.[/b] Bisky adds one to her favorite number. She then divides the result by $2$, and gets $56$. What is her favorite number? [b]p3.[/b] What is the maximum number of points at which a circle and a square can intersect? [b]p4.[/b] An integer $N$ leaves a remainder of 66 when divided by $120$. Find the remainder when $N$ is divided by $24$. [b]p5.[/b] $7$ people are chosen to run for student council. How many ways are there to pick $1$ president, $1$ vice president, and $1$ secretary? [b]p6.[/b] Anya, Beth, Chloe, and Dmitri are all close friends, and like to make group chats to talk. How many group chats can be made if Dmitri, the gossip, must always be in the group chat and Anya is never included in them? Group chats must have more than one person. [b]p7.[/b] There exists a telephone pole of height $24$ feet. From the top of this pole, there are two wires reaching the ground in opposite directions, with one wire $25$ feet, and the other wire 40 feet. What is the distance (in feet) between the places where the wires hit the ground? [b]p8.[/b] Tarik is dressing up for a job-interview. He can wear a chill, business, or casual outfit. If he wears a chill oufit, he must wear a t-shirt, shorts, and flip-flops. He has eight of the first, seven of the second, and three of the third. If he wears a business outfit, he must wear a blazer, a tie, and khakis; he has two of the first, six of the second, and five of the third; finally, he can also choose the casual style, for which he has three hoodies, nine jeans, and two pairs of sneakers. How many different combinations are there for his interview? [b]p9.[/b] If a non-degenerate triangle has sides $11$ and $13$, what is the sum of all possibilities for the third side length, given that the third side has integral length? [b]p10.[/b] An unknown disease is spreading fast. For every person who has the this illness, it is spread on to $3$ new people each day. If Mary is the only person with this illness at the start of Monday, how many people will have contracted the illness at the end of Thursday? [b]p11.[/b] Gob the giant takes a walk around the equator on Mars, completing one lap around Mars. If Gob’s head is $\frac{13}{\pi}$ meters above his feet, how much farther (in meters) did his head travel than his feet? [b]p12.[/b] $2022$ leaves a remainder of $2$, $6$, $9$, and $7$ when divided by $4$, $7$, $11$, and $13$ respectively. What is the next positive integer which has the same remainders to these divisors? [b]p13.[/b] In triangle $ABC$, $AB = 20$, $BC = 21$, and $AC = 29$. Let D be a point on $AC$ such that $\angle ABD = 45^o$. If the length of $AD$ can be represented as $\frac{a}{b}$ , what is $a + b$? [b]p14.[/b] Find the number of primes less than $100$ such that when $1$ is added to the prime, the resulting number has $3$ divisors. [b]p15.[/b] What is the coefficient of the term $a^4z^3$ in the expanded form of $(z - 2a)^7$? [b]p16.[/b] Let $\ell$ and $m$ be lines with slopes $-2$, $1$ respectively. Compute $|s_1 \cdot s_2|$ if $s_1$, $s_2$ represent the slopes of the two distinct angle bisectors of $\ell$ and $m$. [b]p17.[/b] R1D2, Lord Byron, and Ryon are creatures from various planets. They are collecting monkeys for King Avanish, who only understands octal (base $8$). R1D2 only understands binary (base $2$), Lord Byron only understands quarternary (base $4$), and Ryon only understands decimal (base $10$). R1D2 says he has $101010101$ monkeys and adds his monkey to the pile. Lord Byron says he has $3231$ monkeys and adds them to the pile. Ryon says he has $576$ monkeys and adds them to the pile. If King Avanish says he has $x$ monkeys, what is the value of $x$? [b]p18.[/b] A quadrilateral is defined by the origin, $(3, 0)$, $(0, 10)$, and the vertex of the graph of $y = x^2 -8x+22$. What is the area of this quadrilateral? [b]p19.[/b] There is a sphere-container, filled to the brim with fruit punch, of diameter $6$. The contents of this container are poured into a rectangular prism container, again filled to the brim, of dimensions $2\pi$ by $4$ by $3$. However, there is an excess amount in the original container. If all the excess drink is poured into conical containers with diameter $4$ and height $3$, how many containers will be used? [b]p20.[/b] Brian is shooting arrows at a target, made of concurrent circles of radius $1$, $2$, $3$, and $4$. He gets $10$ points for hitting the innermost circle, $8$ for hitting between the smallest and second smallest circles, $5$ for between the second and third smallest circles, $2$ points for between the third smallest and outermost circle, and no points for missing the target. Assume for each shot he takes, there is a $20\%$ chance Brian will miss the target, but otherwise the chances of hitting each target are proportional to the area of the region. The chance that after three shots, Brian will have scored $15$ points can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $m + n$. [b]p21.[/b] What is the largest possible integer value of $n$ such that $\frac{2n^3+n^2+7n-15}{2n+1}$ is an integer? [b]p22.[/b] Let $f(x, y) = x^3 + x^2y + xy^2 + y^3$. Compute $f(0, 2) + f(1, 3) +... f(9, 11).$ [b]p23.[/b] Let $\vartriangle ABC$ be a triangle. Let $AM$ be a median from $A$. Let the perpendicular bisector of segment $\overline{AM}$ meet $AB$ and $AC$ at $D$, $E$ respectively. Given that $AE = 7$, $ME = MC$, and $BDEC$ is cyclic, then compute $AM^2$. [b]p24.[/b] Compute the number of ordered triples of positive integers $(a, b, c)$ such that $a \le 10$, $b \le 11$, $c \le 12$ and $a > b - 1$ and $b > c - 1$. [b]p25.[/b] For a positive integer $n$, denote by $\sigma (n)$ the the sum of the positive integer divisors of $n$. Given that $n + \sigma (n)$ is odd, how many possible values of $n$ are there from $1$ to $2022$, inclusive? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Peter wants to make an unusual die having different positive integers on each of its faces. For neighbouring faces the corresponding numbers should differ by at least two. Find the minimal sum of the six numbers. (Folklore)

We are given $64$ cubes, each with five white faces and one black face. One cube is placed on each square of a chessboard, with its edges parallel to the sides of the board. We are allowed to rotate a complete row of cubes about the axis of symmetry running through the cubes or to rotate a complete column of cubes about the axis of symmetry running through the cubes. Show that by a sequence of such rotations we can always arrange that each cube has its black face uppermost

Let $a, b, c$ be nonzero integers, with $1$ as their only positive common divisor, such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}= 0$. Find the number of such triples $(a, b, c)$ with $50 \ge |a| \ge |b| \ge |c| 1$.

Initially, there are 2018 distinct boxes on a table. In the first stage, Yazan and Bozan, starting with Yazan, take turns make $2016$ moves each, such that, in each move, the person whose turn selects a pair of boxes that is not written on the board, and writes the pair on the board. In the second stage, Bozan enumerates the $4032$ pairs with numbers from $1,2,\dots,4032$, in whichever order he wants, and puts $k$ balls in each boxes written contained in the $k^{th}$ pair. Is there a strategy for Bozan that guarantees that the number of balls in each box are distinct?

Using a nozzle to paint each square in a $1 \times n$ stripe, when the nozzle is aiming at the $i$-th square, the square is painted black, and simultaneously, its left and right neighboring square (if exists) each has an independent probability of $\tfrac{1}{2}$ to be painted black. In the optimal strategy (i.e. achieving least possible number of painting), the expectation of number of painting to paint all the squares black, is $T(n)$. Find the explicit formula of $T(n)$.

There is $n\times n$ chessboard. Each square has a number between $0$ and $k$. There is a button for each row and column, which increases the number of $n$ numbers of the row or column the button represents(if the number of the square is $k$, then it becomes $0$). If certain button is pressed, call it 'operation.' And we have a chessboard which is filled with 0(for all squares). After some 'operation's, the numbers of squares are different now. Prove that we can make all of the number $0$ within $kn$ 'operation's.

At Port Aventura there are $16$ secret agents, each of whom is watching one or more other agents. It is known that if agent $A$ is watching agent $B$, then $B$ is not watching $A$. Moreover, any $10$ agents can be ordered so that the first is watching the second, the second is watching the third, etc, the last is watching the first. Show that any $11$ agents can also be so ordered.

On a Cartesian plane 101 planes are drawn and all points of intersection are labeled. Is it possible, that for every line, 50 of the points have positive coordinates and 50 of the points have negative coordinates [I]Proposed by S. Ivanov[/i]

Two countries ($A$ and $B$) organize a conference, and they send an equal number of participants. Some of them have known each other from a previous conference. Prove that one can choose a nonempty subset $C$ of the participants from $A$ such that one of the following holds: [list][*]the participants from $B$ each know an even number of people in $C$, [*]the participants from $B$ each know an odd number of participants in $C$.[/list]

A bank has one million clients, one of whom is Inspector Gadget. Each client has a unique PIN number consisting of six digits. Dr. Claw has a list of all the clients. He is able to break into the account of any client, choose any $n$ digits of the PIN number and copy them. The n digits he copies from different clients need not be in the same $n$ positions. He can break into the account of each client, but only once. What is the smallest value of $n$ which allows Dr.Claw to determine the complete PIN number of Inspector Gadget?

Let $P$ be a set of $n \ge 2$ distinct points in the plane, which does not contain any triplet of aligned points. Let $S$ be the set of all segments whose endpoints are points of $P$. Given two segments $s_1, s_2 \in S$, we write $s_1 \otimes s_2$ if the intersection of $s_1$ with $s_2$ is a point other than the endpoints of $s_1$ and $s_2$. Prove that there exists a segment $s_0 \in S$ such that the set $\{s \in S | s_0 \otimes s\}$ has at least $\frac{1}{15}\dbinom{n-2}{2}$ elements

For each natural $n \ge 4$, find the smallest natural number $k$ that satisfies following condition: For an arbitrary arrangement of $k$ chips of two colors on $n\times n$ board, there exists a non-empty set such that all columns and rows contain even number ($0$ is also possible) of chips each color.

Four sages stand around a non-transparent baobab. Each of the sages wears red, blue, or green hat. A sage sees only his two neighbors. Each of them at the same time must make a guess about the color of his hat. If at least one sage guesses correctly, the sages win. They could consult before the game started. How should they act to win?

Initially, the numbers $1,1,-1$ written on the board.At every step,Mikail chooses the two numbers $a,b$ and substitutes them with $2a+c$ and $\frac{b-c}{2}$ where $c$ is the unchosen number on the board. Prove that at least $1$ negative number must be remained on the board at any step.