Let $n\geqslant 2$ be a positive integer. Paul has a $1\times n^2$ rectangular strip consisting of $n^2$ unit squares, where the $i^{\text{th}}$ square is labelled with $i$ for all $1\leqslant i\leqslant n^2$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then [i]translate[/i] (without rotating or flipping) the pieces to obtain an $n\times n$ square satisfying the following property: if the unit square in the $i^{\text{th}}$ row and $j^{\text{th}}$ column is labelled with $a_{ij}$, then $a_{ij}-(i+j-1)$ is divisible by $n$. Determine the smallest number of pieces Paul needs to make in order to accomplish this.

A certain town is represented as an infinite plane, which is divided by straight lines into squares. The lines are streets, while the squares are blocks. Along a certain street there stands a policeman on each $100$th intersection . Somewhere in the town there is a bandit , whose position and speed are unknown, but he can move only along the streets. The aim of the police is to see the bandit . Does there exist an algorithm available to the police to enable them to achieve their aim? (A. Andjans, Riga)

Consider a table with $n$ rows and $2n$ columns. we put some blocks in some of the cells. After putting blocks in the table we put a robot on a cell and it starts moving in one of the directions right, left, down or up. It can change the direction only when it reaches a block or border. Find the smallest number $m$ such that we can put $m$ blocks on the table and choose a starting point for the robot so it can visit all of the unblocked cells. (the robot can't enter the blocked cells.) Proposed by Seyed Mohammad Seyedjavadi and Alireza Tavakoli

21 players participated in a tennis tournament, in which each pair of players played exactly once and each game had a winner (no ties are allowed). The organizers of the tournament found out that each player won at least 9 games, and lost at least 9. In addition, they discovered cases of three players $A,B,C$ in which $A$ won against $B$, $B$ won against $C$ and $C$ won against $A$, and called such triples "problematic". [b]a)[/b] What is the maximum possible number of problematic triples? [b]b)[/b] What is the minimum possible number of problematic triples?

[b]p1[/b]. Evaluate $1! + 2! + 3! + 4! + 5! $ (where $n!$ is the product of all integers from $1$ to $n$, inclusive). [b]p2.[/b] Harold opens a pack of Bertie Bott's Every Flavor Beans that contains $10$ blueberry, $10$ watermelon, $3$ spinach and $2$ earwax-flavored jelly beans. If he picks a jelly bean at random, then what is the probability that it is not spinach-flavored? [b]p3.[/b] Find the sum of the positive factors of $32$ (including $32$ itself). [b]p4.[/b] Carol stands at a flag pole that is $21$ feet tall. She begins to walk in the direction of the flag's shadow to say hi to her friends. When she has walked $10$ feet, her shadow passes the flag's shadow. Given that Carol is exactly $5$ feet tall, how long in feet is her shadow? [b]p5.[/b] A solid metal sphere of radius $7$ cm is melted and reshaped into four solid metal spheres with radii $1$, $5$, $6$, and $x$ cm. What is the value of $x$? [b]p6.[/b] Let $A = (2,-2)$ and $B = (-3, 3)$. If $(a,0)$ and $(0, b)$ are both equidistant from $A$ and $B$, then what is the value of $a + b$? [b]p7.[/b] For every flip, there is an $x^2$ percent chance of flipping heads, where $x$ is the number of flips that have already been made. What is the probability that my first three flips will all come up tails? [b]p8.[/b] Consider the sequence of letters $Z\,\,W\,\,Y\,\,X\,\,V$. There are two ways to modify the sequence: we can either swap two adjacent letters or reverse the entire sequence. What is the least number of these changes we need to make in order to put the letters in alphabetical order? [b]p9.[/b] A square and a rectangle overlap each other such that the area inside the square but outside the rectangle is equal to the area inside the rectangle but outside the square. If the area of the rectangle is $169$, then find the side length of the square. [b]p10.[/b] If $A = 50\sqrt3$, $B = 60\sqrt2$, and $C = 85$, then order $A$, $B$, and $C$ from least to greatest. [b]p11.[/b] How many ways are there to arrange the letters of the word $RACECAR$? (Identical letters are assumed to be indistinguishable.) [b]p12.[/b] A cube and a regular tetrahedron (which has four faces composed of equilateral triangles) have the same surface area. Let $r$ be the ratio of the edge length of the cube to the edge length of the tetrahedron. Find $r^2$. [b]p13.[/b] Given that $x^2 + x + \frac{1}{x} +\frac{1}{x^2} = 10$, find all possible values of $x +\frac{1}{x}$ . [b]p14.[/b] Astronaut Bob has a rope one unit long. He must attach one end to his spacesuit and one end to his stationary spacecraft, which assumes the shape of a box with dimensions $3\times 2\times 2$. If he can attach and re-attach the rope onto any point on the surface of his spacecraft, then what is the total volume of space outside of the spacecraft that Bob can reach? Assume that Bob's size is negligible. [b]p15.[/b] Triangle $ABC$ has $AB = 4$, $BC = 3$, and $AC = 5$. Point $B$ is reflected across $\overline{AC}$ to point $B'$. The lines that contain $AB'$ and $BC$ are then drawn to intersect at point $D$. Find $AD$. [b]p16.[/b] Consider a rectangle $ABCD$ with side lengths $5$ and $12$. If a circle tangent to all sides of $\vartriangle ABD$ and a circle tangent to all sides of $\vartriangle BCD$ are drawn, then how far apart are the centers of the circles? [b]p17.[/b] An increasing geometric sequence $a_0, a_1, a_2,...$ has a positive common ratio. Also, the value of $a_3 + a_2 - a_1 - a_0$ is equal to half the value of $a_4 - a_0$. What is the value of the common ratio? [b]p18.[/b] In triangle $ABC$, $AB = 9$, $BC = 11$, and $AC = 16$. Points $E$ and $F$ are on $\overline{AB}$ and $\overline{BC}$, respectively, such that $BE = BF = 4$. What is the area of triangle $CEF$? [b]p19.[/b] Xavier, Yuna, and Zach are running around a circular track. The three start at one point and run clockwise, each at a constant speed. After $8$ minutes, Zach passes Xavier for the first time. Xavier first passes Yuna for the first time in $12$ minutes. After how many seconds since the three began running did Zach first pass Yuna? [b]p20.[/b] How many unit fractions are there such that their decimal equivalent has a cycle of $6$ repeating integers? Exclude fractions that repeat in cycles of $1$, $2$, or $3$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p$ be an odd prime, and put $N=\frac{1}{4} (p^3 -p) -1.$ The numbers $1,2, \dots, N$ are painted arbitrarily in two colors, red and blue. For any positive integer $n \leqslant N,$ denote $r(n)$ the fraction of integers $\{ 1,2, \dots, n \}$ that are red. Prove that there exists a positive integer $a \in \{ 1,2, \dots, p-1\}$ such that $r(n) \neq a/p$ for all $n = 1,2, \dots , N.$ [I]Netherlands[/i]

[b]p1.[/b] A number of Exonians took a math test. If all of their scores were positive integers and the mean of their scores was $8.6$, find the minimum possible number of students. [b]p2.[/b] Find the least composite positive integer that is not divisible by any of $3, 4$, and $5$. [b]p3.[/b] Five checkers are on the squares of an $8\times 8$ checkerboard such that no two checkers are in the same row or the same column. How many squares on the checkerboard share neither a row nor a column with any of the five checkers? [b]p4.[/b] Let the operation $x@y$ be $y - x$. Compute $((... ((1@2)@3)@ ...@ 2013)@2014)@2015$. [b]p5.[/b] In a town, each family has either one or two children. According to a recent survey, $40\%$ of the children in the town have a sibling. What fraction of the families in the town have two children? [b]p6.[/b] Equilateral triangles $ABE$, $BCF$, $CDG$ and $DAH$ are constructed outside the unit square $ABCD$. Eliza wants to stand inside octagon $AEBFCGDH$ so that she can see every point in the octagon without being blocked by an edge. What is the area of the region in which she can stand? [b]p7.[/b] Let $S$ be the string $0101010101010$. Determine the number of substrings containing an odd number of $1$'s. (A substring is defined by a pair of (not necessarily distinct) characters of the string and represents the characters between, inclusively, the two elements of the string.) [b]p8.[/b] Let the positive divisors of $n$ be $d_1, d_2, ...$ in increasing order. If $d_6 = 35$, determine the minimum possible value of $n$. [b]p9.[/b] The unit squares on the coordinate plane that have four lattice point vertices are colored black or white, as on a chessboard, shown on the diagram below. [img]https://cdn.artofproblemsolving.com/attachments/6/4/f400d52ae9e8131cacb90b2de942a48662ea8c.png[/img] For an ordered pair $(m, n)$, let $OXZY$ be the rectangle with vertices $O = (0, 0)$, $X = (m, 0)$, $Z = (m, n)$ and $Y = (0, n)$. How many ordered pairs $(m, n)$ of nonzero integers exist such that rectangle $OXZY$ contains exactly $32$ black squares? [b]p10.[/b] In triangle $ABC$, $AB = 2BC$. Given that $M$ is the midpoint of $AB$ and $\angle MCA = 60^o$, compute $\frac{CM}{AC}$ . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose that $n$ teterahedra are given in space such that any two of them have at least two common vertices, but any three have at most one common vertex. Find the greatest possible $n$.

A few people came to the party. Prove that they can be placed in two rooms so that each of them has in their own room an even number of acquaintances. (One of the rooms can be left empty.)

Anna and Orjan play the following game: they start with a positive integer $n>1$, Anna writes it as the sum of two other positive integers, $n = n_1+n_2$. Orjan deletes one of them, $n_1$ or $n_2$. If the remaining number is larger than $1$, the process is repeated, i.e. Anna writes it as the sum of two positive integers, $ n_3+n_4$, Orjan deletes one of them etc. The game ends when the last number is $1$. Orjan is the winner if there are two equal numbers among the numbers he has deleted, otherwise Anna wins. Who is winning the game if n = 2008 and they both play optimally?

$A_1, A_2, \ldots, A_{29}$ are $29$ different sequences of positive integers. For $1 \leq i < j \leq 29$ and any natural number $x,$ we define $N_i(x) =$ number of elements of the sequence $A_i$ which are less or equal to $x,$ and $N_{ij}(x) =$ number of elements of the intersection $A_i \cap A_j$ which are less than or equal to $x.$ It is given for all $1 \leq i \leq 29$ and every natural number $x,$ \[ N_i(x) \geq \frac{x}{e}, \] where $e = 2.71828 \ldots$ Prove that there exist at least one pair $i,j$ ($1 \leq i < j \leq 29$) such that \[ N_{ij}(1988) > 200. \]

Nestor ordered Juan to do the following work: draw a circle, draw one of its diameters and mark the extreme points of the diameter with the numbers 1 and 2 respectively. Place 100 points in each of the semicircles that determines the diameter layout (different from the ends of the diameter) and mark these points randomly with the numbers $1$ and $2$. To finish, paint red all small segments that have different markings on their extremes. After a certain amount of time passed, Juan finished the work and told Nestor that “he painted 47 segments red.” Prove that if Juan made no mistakes, what he said is false.

Let $n$ be a positive integer. Initially the sequence $0,0,\cdots,0$ ($n$ times) is written on the board. In each round, Ananya choses an integer $t$ and a subset of the numbers written on the board and adds $t$ to all of them. What is the minimum number of rounds in which Ananya can make the sequence on the board strictly increasing? Proposed by Shantanu Nene

For a table $n \times 9$ ($n$ rows and $9$ columns), determine the maximum of $n$ that we can write one number in the set $\left\{ {1,2,...,9} \right\}$ in each cell such that these conditions are satisfied: 1. Each row contains enough $9$ numbers of the set $\left\{ {1,2,...,9} \right\}$. 2. Any two rows are distinct. 3. For any two rows, we can find at least one column such that the two intersecting cells between it and the two rows contain the same number.

Is it possible to divide a polygon with $21$ sides into $2022$ triangles in such a way that among all the vertices there are not three collinear?

Let $n\geq 3$ be an integer. We say that a vertex $A_i (1\leq i\leq n)$ of a convex polygon $A_1A_2 \dots A_n$ is [i]Bohemian[/i] if its reflection with respect to the midpoint of $A_{i-1}A_{i+1}$ (with $A_0=A_n$ and $A_1=A_{n+1}$) lies inside or on the boundary of the polygon $A_1A_2\dots A_n$. Determine the smallest possible number of Bohemian vertices a convex $n$-gon can have (depending on $n$). [i]Proposed by Dominik Burek, Poland [/i]

An integer point of the usual Euclidean $3$-dimensional space is a point whose three coordinates are all integers. A set $S$ of integer points is called a [i]covered [/i] set if for all points $A, B$ in $S$ each integer point in the segment $[AB]$ is also in $S$. Determine the maximum number of elements that a covered set can have if it does not contain $2004$ collinear points.

The Imomi archipelago consists of $n\geq 2$ islands. Between each pair of distinct islands is a unique ferry line that runs in both directions, and each ferry line is operated by one of $k$ companies. It is known that if any one of the $k$ companies closes all its ferry lines, then it becomes impossible for a traveller, no matter where the traveller starts at, to visit all the islands exactly once (in particular, not returning to the island the traveller started at). Determine the maximal possible value of $k$ in terms of $n$. [i]Anton Trygub, Ukraine[/i]

[b]p1.[/b] Insert "$+$" signs between some of the digits in the following sequence to obtain correct equality: $$1\,\,\,\, 2\,\,\,\, 3\,\,\,\, 4\,\,\,\,5\,\,\,\, 6\,\,\,\, 7 = 100$$ [b]p2.[/b] A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the big square $ABCD$ is $40$ cm. [img]https://cdn.artofproblemsolving.com/attachments/8/c/d54925cba07f63ec8578048f46e1e730cb8df3.png[/img] [b]p3.[/b] Jack made $3$ quarts of fruit drink from orange and apple juice. $\frac25$ of his drink is orange juice and the rest is apple juice. Nick prefers more orange juice in the drink. How much orange juice should he add to the drink to obtain a drink composed of $\frac35$ of orange juice? [b]p4.[/b] A train moving at $55$ miles per hour meets and is passed by a train moving moving in the opposite direction at $35$ miles per hour. A passenger in the first train sees that the second train takes $8$ seconds to pass him. How long is the second train? [b]p5.[/b] It is easy to arrange $16$ checkers in $10$ rows of $4$ checkers each, but harder to arrange $9$ checkers in $10$ rows of $3$ checkers each. Do both. [b]p6.[/b] Every human that lived on Earth exchanged some number of handshakes with other humans. Show that the number of people that made an odd number of handshakes is even. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the number of distinguishable groupings into which you can place $3$ indistinguishable red balls and $3$ indistinguishable blue balls. Here the groupings $RR-BR-B-B$ and $B-RB-B-RR$ are indistinguishable because the groupings are merely rearranged, but $RRB-BR-B$ is distinguishable from $RBB-BR-R$.

Alice and Bob play a game on a connected graph with $2n$ vertices, where $n\in \mathbb{N}$ and $n>1$.. Alice and Bob have tokens named A and B respectively. They alternate their turns with Alice going first. Alice gets to decide the starting positions of A and B. Every move, the player with the turn moves their token to an adjacent vertex. Bob's goal is to catch Alice, and Alice's goal is to prevent this. Note that positions of A, B are visible to both Alice and Bob at every moment. Provided they both play optimally, what is the maximum possible number of edges in the graph if Alice is able to evade Bob indefinitely? [i]Proposed by Shashank Ingalagavi and Vighnesh Sangle[/i]

Fix an integer $n \geq 2$. An $n\times n$ sieve is an $n\times n$ array with $n$ cells removed so that exactly one cell is removed from every row and every column. A stick is a $1\times k$ or $k\times 1$ array for any positive integer $k$. For any sieve $A$, let $m(A)$ be the minimal number of sticks required to partition $A$. Find all possible values of $m(A)$, as $A$ varies over all possible $n\times n$ sieves. [i]Palmer Mebane[/i]

[b]p1.[/b] What is $6\times 7 + 4 \times 7 + 6\times 3 + 4\times 3$? [b]p2.[/b] How many integers $n$ have exactly $\sqrt{n}$ factors? [b]p3.[/b] A triangle has distinct angles $3x+10$, $2x+20$, and $x+30$. What is the value of $x$? [b]p4.[/b] If $4$ people of the Math Club are randomly chosen to be captains, and Henry is one of the $30$ people eligible to be chosen, what is the probability that he is not chosen to be captain? [b]p5.[/b] $a, b, c, d$ is an arithmetic sequence with difference $x$ such that $a, c, d$ is a geometric sequence. If $b$ is $12$, what is $x$? (Note: the difference of an aritmetic sequence can be positive or negative, but not $0$) [b]p6.[/b] What is the smallest positive integer that contains only $0$s and $5$s that is a multiple of $24$. [b]p7.[/b] If $ABC$ is a triangle with side lengths $13$, $14$, and $15$, what is the area of the triangle made by connecting the points at the midpoints of its sides? [b]p8.[/b] How many ways are there to order the numbers $1,2,3,4,5,6,7,8$ such that $1$ and $8$ are not adjacent? [b]p9.[/b] Find all ordered triples of nonnegative integers $(x, y, z)$ such that $x + y + z = xyz$. [b]p10.[/b] Noah inscribes equilateral triangle $ABC$ with area $\sqrt3$ in a cricle. If $BR$ is a diameter of the circle, then what is the arc length of Noah's $ARC$? [b]p11.[/b] Today, $4/12/14$, is a palindromic date, because the number without slashes $41214$ is a palindrome. What is the last palindromic date before the year $3000$? [b]p12.[/b] Every other vertex of a regular hexagon is connected to form an equilateral triangle. What is the ratio of the area of the triangle to that of the hexagon? [b]p13.[/b] How many ways are there to pick four cards from a deck, none of which are the same suit or number as another, if order is not important? [b]p14.[/b] Find all functions $f$ from $R \to R$ such that $f(x + y) + f(x - y) = x^2 + y^2$. [b]p15.[/b] What are the last four digits of $1(1!) + 2(2!) + 3(3!) + ... + 2013(2013!)$/ [b]p16.[/b] In how many distinct ways can a regular octagon be divided up into $6$ non-overlapping triangles? [b]p17.[/b] Find the sum of the solutions to the equation $\frac{1}{x-3} + \frac{1}{x-5} + \frac{1}{x-7} + \frac{1}{x-9} = 2014$ . [b]p18.[/b] How many integers $n$ have the property that $(n+1)(n+2)(n+3)(n+4)$ is a perfect square of an integer? [b]p19.[/b] A quadrilateral is inscribed in a unit circle, and another one is circumscribed. What is the minimum possible area in between the two quadrilaterals? [b]p20.[/b] In blindfolded solitary tic-tac-toe, a player starts with a blank $3$-by-$3$ tic-tac-toe board. On each turn, he randomly places an "$X$" in one of the open spaces on the board. The game ends when the player gets $3$ $X$s in a row, in a column, or in a diagonal as per normal tic-tac-toe rules. (Note that only $X$s are used, not $O$s). What fraction of games will run the maximum $7$ amount of moves? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A teacher and her 30 students play a game on an infinite cell grid. The teacher starts first, then each of the 30 students makes a move, then the teacher and so on. On one move the person can color one unit segment on the grid. A segment cannot be colored twice. The teacher wins if, after the move of one of the 31 players, there is a $1\times 2$ or $2\times 1$ rectangle , such that each segment from it's border is colored, but the segment between the two adjacent squares isn't colored. Prove that the teacher can win.

Students in a class of $n$ students had to solve $2^{n-1}$ problems on an exam. It turned out that for each pair of distinct problems: • there is at least one student who has solved both • there is at least one student who has solved one of them, but not the other. Show that there is a problem solved by all the students in the class.