The 16 fields of a $4 \times 4$ checker board can be arranged in 18 lines as follows: the four lines, the four columns, the five diagonals from north west to south east and the five diagonals from north east to south west. These diagonals consists of 2,3 or 4 edge-adjacent fields of same colour; the corner fields of the chess board alone do not form a diagonal. Now, we put a token in 10 of the 16 fields. Each of the 18 lines contains an even number of tokens contains a point. What is the highest possible point number when can be achieved by optimal placing of the 10 tokens. Explain your answer.

The numbers $1$ through $9$ are written on a $3 \times 3$ board, without repetitions. A valid operation is to choose a row or a column of the board, and replace its three numbers $a, b, c$ (in order, i.e., the first number of the row/column is $a$, the second number of the row/column is $b$, the third number of the row/column is $c$) with either the three non-negative numbers $a-x, b-x, c+x$ (in order) or with the three non-negative numbers $a+x, b-x, c-x$ (in order), where $x$ is a real positive number which may vary in each operation . a) Determine if there is a way of getting all $9$ numbers on the board to be the same, starting with the following board: $\begin{array}{|c|c|c|c|c|c|c|c|} \hline 1 & 2 & 3 \\ \hline 4 & 5 & 6 \\ \hline 7 & 8 & 9 \\ \hline \end{array}$ b) For all posible configurations such that it is possible to get all $9$ numbers to be equal to a number $m$ using the valid operations, determine the maximum value of $m$.

Alice and Bob each flip $20$ fair coins. Given that Alice flipped at least as many heads as Bob, what is the expected number of heads that Alice flipped? [i]Proposed by Adam Bertelli[/i]

Let $A$ and $B$ be two finite disjoint sets of points in the plane such that no three distinct points in $A \cup B$ are collinear. Assume that at least one of the sets $A, B$ contains at least five points. Show that there exists a triangle all of whose vertices are contained in $A$ or in $B$ that does not contain in its interior any point from the other set.

Let $m$, $n$ be two positive integers greater than 3. Consider the table of size $m\times n$ ($m$ rows and $n$ columns) formed with unit squares. We are putting marbles into unit squares of the table following the instructions: $-$ each time put 4 marbles into 4 unit squares (1 marble per square) such that the 4 unit squares formes one of the followings 4 pictures (click [url=http://www.mathlinks.ro/Forum/download.php?id=4425]here[/url] to view the pictures). In each of the following cases, answer with justification to the following question: Is it possible that after a finite number of steps we can set the marbles into all of the unit squares such that the numbers of marbles in each unit square is the same? a) $m=2004$, $n=2006$; b) $m=2005$, $n=2006$.

The captain and his three sailors get $2009$ golden coins with the same value . The four people decided to divide these coins by the following rules : sailor $1$,sailor $2$,sailor $3$ everyone write down an integer $b_1,b_2,b_3$ , satisfies $b_1\ge b_2\ge b_3$ , and ${b_1+b_2+b_3=2009}$; the captain dosen't know what the numbers the sailors have written . He divides $2009$ coins into $3$ piles , with number of coins: $a_1,a_2,a_3$ , and $a_1\ge a_2\ge a_3$ . For sailor $k$ ($k=1,2,3$) , if $b_k<a_k$ , then he can take $b_k$ coins from the $k$th pile ; if $b_k\ge a_k$ , then he can't take any coins away . At last , the captain own the rest of the coins .If no matter what the numbers the sailors write , the captain can make sure that he always gets $n$ coins . Find the largest possible value of $n$ and prove your conclusion .

Prove that we can color every subset with $n$ element of a set with $3n$ elements with $8$ colors . In such a way that there are no $3$ subsets $A,B,C$ with the same color where : $$|A \cap B| \le 1,|A \cap C| \le 1,|B \cap C| \le 1$$ Proposed by [i]Morteza Saghafian[/i] and [i]Amir Jafari[/i]

In a plane we have $n$ lines, no two of which are parallel or perpendicular, and no three of which are concurrent. A cartesian system of coordinates is chosen for the plane with one of the lines as the $x$-axis. A point $P$ is located at the origin of the coordinate system and starts moving along the positive $x$-axis with constant velocity. Whenever $P$ reaches the intersection of two lines, it continues along the line it just reached in the direction that increases its $x$-coordinate. Show that it is possible to choose the system of coordinates in such a way that $P$ visits points from all $n$ lines.

[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].

Martin has two boxes $A$ and $B$. In the box $A$ there are $100$ red balls numbered from $1$ to $100$, each one with one of these numbers. In the box $B$ there are $100$ blue balls numbered from $101$ to $200$, each one with one of these numbers. Martin chooses two positive integers $a$ and $b$, both less than or equal to $100$, and then he takes out $a$ balls from box $A$ and $b$ balls from box $B$, without replacement. Martin's goal is to have two red balls and one blue ball among all balls taken such that the sum of the numbers of two red balls equals the number of the blue ball.\\ What is the least possible value of $a+b$ so that Martin achieves his goal for sure? For such a minimum value of $a+b$, give an example of $a$ and $b$ satisfying the goal and explain why every $a$ and $b$ with smaller sum cannot accomplish the aim.

In a club with $30$ members, every member initially had a hat. One day each member sent his hat to a different member (a member could have received more than one hat). Prove that there exists a group of $10$ members such that no one in the group has received a hat from another one in the group.

[b]p1.[/b] Sherry starts with a three-digit positive integer. She subtracts $7$ from it, then multiplies the result by $7$, and then adds $7$ to that. If she ends up with $2023$, what number did she start with? [b]p2.[/b] Square $ABCD$ has side length $1$. Point $X$ lies on $\overline{AB}$ such that $\frac{AX}{XB} = 2$, and point $Y$ lies on $\overline{DX}$ such that $\frac{DY}{YX} = 3$. Compute the area of triangle $DAY$ . [b]p3.[/b] A fair six-sided die is labeled $1-6$ such that opposite faces sum to $7$. The die is rolled, but before you can look at the outcome, the die gets tipped over to an adjacent face. If the new face shows a $4$, what is the probability the original roll was a $1$? PS. You should use hide for answers.

There is house with $2^n$ rooms and every room has one light bulb and light switch. But wiring was connected wrong, so light switch can turn on light in some another room. Master want to find what switch connected to every light bulb. He use next practice: he send some workers in the some rooms, then they turn on switches in same time, then they go to master and tell him, in what rooms light bulb was turned on. a) Prove that $2n$ moves is enough to find, how switches are connected to bulbs. b) Is $2n-1$ moves always enough ?

Prove that it is not possible to write numbers $ 1,2,3,...,25$ on the squares of $ 5$x$ 5$ chessboard such that any neighboring numbers differ by at most $ 4$.

Karakade has three flash drives of each of the six capacities $1, 2, 4, 8, 16, 32$ gigabytes. She gives each of her $6$ servants three flash drives of different capacities. Prove that either there are two capacities where each servant has at most one of the two capacities, or all servants have flash drives with different sums of capacities.

Bill Gates and Jeff Bezos are playing a game. Each turn, a coin is flipped, and if Bill and Jeff have $m,n>0$ dollars, respectively, the winner of the coin toss will take $\min{(m,n)}$ from the loser. Given that Bill starts with $20$ dollars and Jeff starts with $21$ dollars, what is the probability that Bill ends up with all of the money? [i]Proposed by Daniel Li[/i]

Given natural numbers $p < k < n$. On an endless checkered plane some cells are marked so that in any rectangle $(k + 1) \times n$ ($n$ cells horizontally, $k + 1$ vertically) marked exactly $p$ cells. Prove that there is a $k \times (n + 1)$ rectangle ($n + 1$ cell horizontally, $k$ - vertically), in which no less than $p + 1$ cells.

In some galaxy there are $1000000$ planets and on each of them there are at least $101$ portals. Each portal allows to teleport between some two planets, no two planets are connected by more than one portal. It is known that starting from any planet using portals you can get to any other planet, while it is impossible to return to that planet using once at most 5 different portals. Prove that starting from any planet you can get to any other planet within a year, using at most one portal daily (a year consists of 365 days). [i]M. Zorka[/i]

Three triangles $A_1A_2A_3$, $B_1B_2B_3$, $C_1C_2C_3$ are given such that their centres of gravity (intersection points of their medians) lie on a straight line, but no three of the $9$ vertices of the triangles lie on a straight line. Consider the set of $27$ triangles $A_iB_jC_k$ (where $i$, $j$, $k$ take the values $1$, $2$, $3$ independently). Prove that this set of triangles can be divided into two parts of the same total area. (A. Andjans, Riga)

Does there exist a piecewise linear continuous function $f:\mathbb{R}\to \mathbb{R}$ such that for any two-way infinite sequence $a_n\in[0,1]$, $n\in\mathbb{Z}$, there exists an $x\in\mathbb{R}$ with \[ \limsup_{K\to \infty} \frac{\#\{k\le K\,:\, k\in\mathbb{N},f^k(x)\in[n,n+1)\}}{K}=a_n \] for all $n\in\mathbb{Z}$, where $f^k=f\circ f\circ \dots\circ f$ stands for the $k$-fold iterate of $f$?

Let $X$ be a non-empty and finite set, $A_1,...,A_k$ $k$ subsets of $X$, satisying: (1) $|A_i|\leq 3,i=1,2,...,k$ (2) Any element of $X$ is an element of at least $4$ sets among $A_1,....,A_k$. Show that one can select $[\frac{3k}{7}] $ sets from $A_1,...,A_k$ such that their union is $X$.

[b]p1.[/b] What is the maximum possible value of $m$ such that there exist $m$ integers $a_1, a_2, ..., a_m$ where all the decimal representations of $a_1!, a_2!, ..., a_m!$ end with the same amount of zeros? [b]p2.[/b] Let $f : R \to R$ be a function such that $f(x) + f(y^2) = f(x^2 + y)$, for all $x, y \in R$. Find the sum of all possible $f(-2017)$. [b]p3. [/b] What is the sum of prime factors of $1000027$? [b]p4.[/b] Let $$\frac{1}{2!} +\frac{2}{3!} + ... +\frac{2016}{2017!} =\frac{n}{m},$$ where $n, m$ are relatively prime. Find $(m - n)$. [b]p5.[/b] Determine the number of ordered pairs of real numbers $(x, y)$ such that $\sqrt[3]{3 - x^3 - y^3} =\sqrt{2 - x^2 - y^2}$ [b]p6.[/b] Triangle $\vartriangle ABC$ has $\angle B = 120^o$, $AB = 1$. Find the largest real number $x$ such that $CA - CB > x$ for all possible triangles $\vartriangle ABC$. [b]p7. [/b]Jung and Remy are playing a game with an unfair coin. The coin has a probability of $p$ where its outcome is heads. Each round, Jung and Remy take turns to flip the coin, starting with Jung in round $ 1$. Whoever gets heads first wins the game. Given that Jung has the probability of $8/15$ , what is the value of $p$? [b]p8.[/b] Consider a circle with $7$ equally spaced points marked on it. Each point is $ 1$ unit distance away from its neighbors and labelled $0,1,2,...,6$ in that order counterclockwise. Feng is to jump around the circle, starting at the point $0$ and making six jumps counterclockwise with distinct lengths $a_1, a_2, ..., a_6$ in a way such that he will land on all other six nonzero points afterwards. Let $s$ denote the maximum value of $a_i$. What is the minimum possible value of $s$? [b]p9. [/b]Justin has a $4 \times 4 \times 4$ colorless cube that is made of $64$ unit-cubes. He then colors $m$ unit-cubes such that none of them belong to the same column or row of the original cube. What is the largest possible value of $m$? [b]p10.[/b] Yikai wants to know Liang’s secret code which is a $6$-digit integer $x$. Furthermore, let $d(n)$ denote the digital sum of a positive integer $n$. For instance, $d(14) = 5$ and $d(3) = 3$. It is given that $$x + d(x) + d(d(x)) + d(d(d(x))) = 999868.$$ Please find $x$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $A=[0,0]$ and $B=[n,n]$. In how many ways can we go from $A$ to $B$, if we always want to go from lattice point to its neighbour (i.e. point with one coordinate the same and one smaller or bigger by one), we never want to visit the same point twice and we want our path to have length $2n+2$? (For example, path $[0,0],[0,1],[-1,1],[-1,2],[0,2],[1,2],[2,2],[2,3],[3,3]$ is one of the paths for $n=3$)

On Mars a basketball team consists of 6 players. The coach of the team Mars can select any line-up of 6 players among 100 candidates. The coach considers some line-ups as [i]appropriate[/i] while the other line-ups are not (there exists at least one appropriate line-up). A set of 5 candidates is called [i]perspective[/i] if one more candidate could be added to it to obtain an appropriate line-up. A candidate is called [i]universal[/i] if he completes each perspective set of 5 candidates (not containing him) upto an appropriate line-up. The coach has selected a line-up of 6 universal candidates. Determine if it follows that this line-up is appropriate.

Find all triplets $(k, m, n)$ of positive integers having the following property: Square with side length $m$ can be divided into several rectangles of size $1\times k$ and a square with side length $n$.