What is the maximal number of crosses than can fit in a $10\times 11$ board without overlapping? Is this problem well-known? [asy] size(4.58cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -3.18, xmax = 1.4, ymin = -0.22, ymax = 3.38; /* image dimensions */ /* draw figures */ draw((-3.,2.)--(1.,2.)); draw((-2.,3.)--(-2.,0.)); draw((-2.,0.)--(-1.,0.)); draw((-1.,0.)--(-1.,3.)); draw((-1.,3.)--(-2.,3.)); draw((-3.,1.)--(1.,1.)); draw((1.,1.)--(1.,2.)); draw((-3.,2.)--(-3.,1.)); draw((0.,2.)--(0.,1.)); draw((-1.,2.)--(-1.,1.)); draw((-2.,2.)--(-2.,1.)); /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]

On the control board of a nuclear station, there are $n$ electric switches ($n > 0$), all in one row. Each switch has two possible positions: up and down. The switches are connected to each other in such a way that, whenever a switch moves down from its upper position, its right neighbour (if it exists) automatically changes position. At the beginning, all switches are down. The operator of the board first changes the position of the leftmost switch once, then the position of the second leftmost switch twice etc., until eventually he changes the position of the rightmost switch n times. How many switches are up after all these operations?

There are $n$ positive integers on the board. We can add only positive integers $c=\frac{a+b}{a-b}$, where $a$ and $b$ are numbers already writted on the board. $a)$ Find minimal value of $n$, such that with adding numbers with described method, we can get any positive integer number written on the board $b)$ For such $n$, find numbers written on the board at the beginning

Find the largest positive integer $n$ ($n \ge 3$), so that there is a convex $n$-gon, the tangent of each interior angle is an integer.

Prove that the vertices of any planar graph can be colored with $3$ colors such that there is no monochromatic cycle.

A number is written in each square of a chessboard, so that each number not on the border is the mean of the $4$ neighboring numbers. Show that if the largest number is $N$, then there is a number equal to $N$ in the border squares.

[u]Round 1[/u] [b]p1.[/b] The temperature inside is $28^o$ F. After the temperature is increased by $5^o$ C, what will the new temperature in Fahrenheit be? [b]p2.[/b] Find the least positive integer value of $n$ such that $\sqrt{2021+n}$ is a perfect square. [b]p3.[/b] A heart consists of a square with two semicircles attached by their diameters as shown in the diagram. Given that one of the semicircles has a diameter of length $10$, then the area of the heart can be written as $a +b\pi$ where $a$ and $b$ are positive integers. Find $a +b$. [img]https://cdn.artofproblemsolving.com/attachments/7/b/d277d9ebad76f288504f0d5273e19df568bc44.png[/img] [u]Round 2[/u] [b]p4.[/b] An $L$-shaped tromino is a group of $3$ blocks (where blocks are squares) arranged in a $L$ shape, as pictured below to the left. How many ways are there to fill a $12$ by $2$ rectangle of blocks (pictured below to the right) with $L$-shaped trominos if the trominos can be rotated or reflected? [img]https://cdn.artofproblemsolving.com/attachments/d/c/cf37cdf9703ae0cd31c38af23b6874fddb3c12.png[/img] [b]p5.[/b] How many permutations of the word $PIKACHU$ are there such that no two vowels are next to each other? [b]p6.[/b] Find the number of primes $n$ such that there exists another prime $p$ such that both $n +p$ and $n-p$ are also prime numbers. [u]Round 3[/u] [b]p7.[/b] Maisy the Bear is at the origin of the Cartesian Plane. WhenMaisy is on the point $(m,n)$ then it can jump to either $(m,n +1)$ or $(m+1,n)$. Let $L(x, y)$ be the number of jumps it takes forMaisy to reach point (x, y). The sum of $L(x, y)$ over all lattice points $(x, y)$ with both coordinates between $0$ and $2020$, inclusive, is denoted as $S$. Find $\frac{S}{2020}$ . [b]p8.[/b] A circle with center $O$ and radius $2$ and a circle with center $P$ and radius $3$ are externally tangent at $A$. Points $B$ and $C$ are on the circle with center $O$ such that $\vartriangle ABC$ is equilateral. Segment $AB$ extends past $B$ to point $D$ and $AC$ extends past $C$ to point $E$ such that $BD = CE = \sqrt3$. The area of $\vartriangle DEP$ can be written as $\frac{a\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers such that $b$ is squarefree and $gcd (a,c) = 1$. Find $a +b +c$. [b]p9.[/b] Find the number of trailing zeroes at the end of $$\prod^{2021}_{i=1}(2021+i -1) = (2021)(2022)...(4041).$$ [u]Round 4[/u] [b]p10.[/b] Let $a, b$, and $c$ be side lengths of a rectangular prism with space diagonal $10$. Find the value of $$(a +b)^2 +(b +c)^2 +(c +a)^2 -(a +b +c)^2.$$ [b]p11.[/b] In a regular heptagon $ABCDEFG$, $\ell$ is a line through $E$ perpendicular to $DE$. There is a point $P$ on $\ell$ outside the heptagon such that $PA = BC$. Find the measure of $\angle EPA$. [b]p12.[/b] Dunan is being "$SUS$". The word "$SUS$" is a palindrome. Find the number of palindromes that can be written using some subset of the letters $\{S, U, S, S, Y, B, A, K, A\}$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3166494p28814284]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166500p28814367]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There were finitely many persons at a party among whom some were friends. Among any $4$ of them there were either $3$ who were all friends among each other or $3$ who weren't friend with each other. Prove that you can separate all the people at the party in two groups in such a way that in the first group everyone is friends with each other and that all the people in the second group are not friends to anyone else in second group. (Friendship is a mutual relation).

Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps. [list=1] [*]select a $2\times 2$ square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square. [/list] Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves. [i]Thanasin Nampaisarn, Thailand[/i]

In a checkered square of size $2021\times 2021$ all cells are initially white. Ivan selects two cells and paints them black. At each step, all the cells that have at least one black neighbor by side are painted black simultaneously. Ivan selects the starting two cells so that the entire square is painted black as fast as possible. How many steps will this take? [i]Ivan Yashchenko[/i]

Let $S=\{1,2,3,\ldots,n\}$, $n$ an odd number. Find the parity of number of permutations $\sigma : S \Rightarrow S$ such that the sequence defined by \[a(i)=|\sigma(i)-i|\] is monotonous.

Let $n_1,n_2, \cdots, n_{26}$ be pairwise distinct positive integers satisfying (1) for each $n_i$, its digits belong to the set $\{1,2\}$; (2) for each $i,j$, $n_i$ can't be obtained from $n_j$ by adding some digits on the right. Find the smallest possible value of $\sum_{i=1}^{26} S(n_i)$, where $S(m)$ denotes the sum of all digits of a positive integer $m$.

How many ways can you mark $8$ squares of an $8\times 8$ chessboard so that no two marked squares are in the same row or column, and none of the four corner squares is marked? (Rotations and reflections are considered different.)

[b]p1.[/b] Frankie the frog likes to hop. On his first hop, he hops $1$ meter. On each successive hop, he hops twice as far as he did on the previous hop. For example, on his second hop, he hops $2$ meters, and on his third hop, he hops $4$ meters. How many meters, in total, has he travelled after $6$ hops? [b]p2.[/b] Anton flips $5$ fair coins. The probability that he gets an odd number of heads can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p3.[/b] April discovers that the quadratic polynomial $x^2 + 5x + 3$ has distinct roots $a$ and $b$. She also discovers that the quadratic polynomial $x^2 + 7x + 4$ has distinct roots $c$ and $d$. Compute $$ac + bc + bd + ad + a + b.$$ [b]p4.[/b] A rectangular picture frame that has a $2$ inch border can exactly fit a $10$ by $7$ inch photo. What is the total area of the frame's border around the photo, in square inches? [b]p5.[/b] Compute the median of the positive divisors of $9999$. [b]p6.[/b] Kaity only eats bread, pizza, and salad for her meals. However, she will refuse to have salad if she had pizza for the meal right before. Given that she eats $3$ meals a day (not necessarily distinct), in how many ways can we arrange her meals for the day? [b]p7.[/b] A triangle has side lengths $3$, $4$, and $x$, and another triangle has side lengths $3$, $4$, and $2x$. Assuming both triangles have positive area, compute the number of possible integer values for $x$. [b]p8.[/b] In the diagram below, the largest circle has radius $30$ and the other two white circles each have a radius of $15$. Compute the radius of the shaded circle. [img]https://cdn.artofproblemsolving.com/attachments/c/1/9eaf1064b2445edb15782278fc9c6efd1440b0.png[/img] [b]p9.[/b] What is the remainder when $2022$ is divided by $9$? [b]p10.[/b] For how many positive integers $x$ less than $2022$ is $x^3 - x^2 + x - 1$ prime? [b]p11.[/b] A sphere and cylinder have the same volume, and both have radius $10$. The height of the cylinder can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p12.[/b] Amanda, Brianna, Chad, and Derrick are playing a game where they pass around a red flag. Two players "interact" whenever one passes the flag to the other. How many different ways can the flag be passed among the players such that (1) each pair of players interacts exactly once, and (2) Amanda both starts and ends the game with the flag? [b]p13.[/b] Compute the value of $$\dfrac{12}{1 + \dfrac{12}{1+ \dfrac{12}{1+...}}}$$ [b]p14.[/b] Compute the sum of all positive integers $a$ such that $a^2 - 505$ is a perfect square. [b]p15.[/b] Alissa, Billy, Charles, Donovan, Eli, Faith, and Gerry each ask Sara a question. Sara must answer exactly $5$ of them, and must choose an order in which to answer the questions. Furthermore, Sara must answer Alissa and Billy's questions. In how many ways can Sara complete this task? [b]p16.[/b] The integers $-x$, $x^2 - 1$, and $x3$ form a non-decreasing arithmetic sequence (in that order). Compute the sum of all possible values of $x^3$. [b]p17.[/b] Moor and his $3$ other friends are trying to split burgers equally, but they will have $2$ left over. If they find another friend to split the burgers with, everyone can get an equal amount. What is the fewest number of burgers that Moor and his friends could have started with? [b]p18.[/b] Consider regular dodecagon $ABCDEFGHIJKL$ below. The ratio of the area of rectangle $AFGL$ to the area of the dodecagon can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/8/3/c38c10a9b2f445faae397d8a7bc4c8d3ed0290.png[/img] [b]p19.[/b] Compute the remainder when $3^{4^{5^6}}$ is divided by $4$. [b]p20.[/b] Fred is located at the middle of a $9$ by $11$ lattice (diagram below). At every second, he randomly moves to a neighboring point (left, right, up, or down), each with probability $1/4$. The probability that he is back at the middle after exactly $4$ seconds can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/7/c/f8e092e60f568ab7b28964d23b2ee02cdba7ad.png[/img] PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $k$ be a positive integer. Prove that one can partition the set $\{ 0,1,2,3, \cdots ,2^{k+1}-1 \}$ into two disdinct subsets $\{ x_1,x_2, \cdots, x_{2k} \}$ and $\{ y_1, y_2, \cdots, y_{2k} \}$ such that $\sum_{i=1}^{2^k} x_i^m =\sum_{i=1}^{2^k} y_i^m$ for all $m \in \{ 1,2, \cdots, k \}$.

$1399$ points and some chords between them is given. $a)$ In every step we can take two chords $RS,PQ$ with a common point other than $P,Q,R,S$ and erase [u]exactly one[/u] of $RS,PQ$ and draw $PS,PR,QS,QR$ let $s$ be the minimum of chords after some steps. Find the maximum of $s$ over all initial positions. $b)$ In every step we can take two chords $RS,PQ$ with a common point other than $P,Q,R,S$ and erase [u]both[/u] of $RS,PQ$ and draw $PS,PR,QS,QR$ let $s$ be the minimum of chords after some steps. Find the maximum of $s$ over all initial positions. [i]Proposed by Afrouz Jabalameli, Abolfazl Asadi[/i]

Given positive integers $m$ and $n \ge m$, determine the largest number of dominoes ($1\times2$ or $2 \times 1$ rectangles) that can be placed on a rectangular board with $m$ rows and $2n$ columns consisting of cells ($1 \times 1$ squares) so that: (i) each domino covers exactly two adjacent cells of the board; (ii) no two dominoes overlap; (iii) no two form a $2 \times 2$ square; and (iv) the bottom row of the board is completely covered by $n$ dominoes.

Akello divides a square up into finitely many white and red rectangles, each (rectangle) with sides parallel to the sides of the parent square. Within each white rectangle, she writes down the value of its width divided by its height, while within each red rectangle, she writes down the value of its height divided by its width. Finally, she calculates $x$, the sum of these numbers. If the total area of the white rectangles equals the total area of the red rectangles, what is the least possible value of $x$ she can get?

There is a village of pile-built dwellings on a lake, set on the gridpoints of an $m \times n$ rectangular grid. Each dwelling is connected by exactly $p$ bridges to some of the neighboring dwellings (diagonal connections are not allowed, two dwellings can be connected by more than one bridge). Determine for which values $m,n, p$ it is possible to place the bridges so that from any dwelling one can reach any other dwelling.

Given natural numbers $x_1,x_2,...,x_n,y_1,y_2,...,y_m$. The following condition is valid: $$(x_1+x_2+...+x_n)=(y_1+y_2+...+y_m)<mn \,\,\,\, (*)$$ Prove that it is possible to delete some terms from (*) (not all and at least one) and to obtain another valid condition.

Let $n,k$ be arbitrary positive integers. We fill the entries of an $n\times k$ array with integers such that all the $n$ rows contain the integers $1,2,\dots,k$ in some order. Add up the numbers in all $k$ columns – let $S$ be the largest of these sums. What is the minimal value of $S$?

Let $m \ge n$ be positive integers. Frieder is given $mn$ posters of Linus with different integer dimensions of $k \times l$ with $1 \ge k \ge m$ and $1 \ge l \ge n$. He must put them all up one by one on his bedroom wall without rotating them. Every time he puts up a poster, he can either put it on an empty spot on the wall or on a spot where it entirely covers a single visible poster and does not overlap any other visible poster. Determine the minimal area of the wall that will be covered by posters.

On the surface of a sphere, a non-intersecting closed curve is drawn. It divides the surface of the sphere in two regions, which are coloured red and blue. Prove that there exist two antipodes of different colours. [i]Note: the curve is colourless.[/i] [i]Proposed by Vlad Spătaru[/i]

[b]p1.[/b] Susan plays a game in which she rolls two fair standard six-sided dice with sides labeled one through six. She wins if the number on one of the dice is three times the number on the other die. If Susan plays this game three times, compute the probability that she wins at least once. [b]p2.[/b] In triangles $\vartriangle ABC$ and $\vartriangle DEF$, $DE = 4AB$, $EF = 4BC$, and $FD = 4CA$. The area of $\vartriangle DEF$ is $360$ units more than the area of $\vartriangle ABC$. Compute the area of $\vartriangle ABC$. [b]p3.[/b] Andy has $2010$ square tiles, each of which has a side length of one unit. He plans to arrange the tiles in an $m\times n$ rectangle, where $mn = 2010$. Compute the sum of the perimeters of all of the different possible rectangles he can make. Two rectangles are considered to be the same if one can be rotated to become the other, so, for instance, a $1\times 2010$ rectangle is considered to be the same as a $2010\times 1$ rectangle. [b]p4.[/b] Let $$S = \log_2 9 \log_3 16 \log_4 25 ... \log_{999} 1000000.$$ Compute the greatest integer less than or equal to $\log_2 S$. [b]p5.[/b] Let $A$ and $B$ be fixed points in the plane with distance $AB = 1$. An ant walks on a straight line from point $A$ to some point $C$ in the plane and notices that the distance from itself to B always decreases at any time during this walk. Compute the area of the region in the plane containing all points where point $C$ could possibly be located. [b]p6.[/b] Lisette notices that $2^{10} = 1024$ and $2^{20} = 1 048 576$. Based on these facts, she claims that every number of the form $2^{10k}$ begins with the digit $1$, where k is a positive integer. Compute the smallest $k$ such that Lisette's claim is false. You may or may not find it helpful to know that $ln 2 \approx 0.69314718$, $ln 5 \approx 1.60943791$, and $log_{10} 2 \approx 0:30103000$. [b]p7.[/b] Let $S$ be the set of all positive integers relatively prime to $6$. Find the value of $\sum_{k\in S}\frac{1}{2^k}$ . [b]p8.[/b] Euclid's algorithm is a way of computing the greatest common divisor of two positive integers $a$ and $b$ with $a > b$. The algorithm works by writing a sequence of pairs of integers as follows. 1. Write down $(a, b)$. 2. Look at the last pair of integers you wrote down, and call it $(c, d)$. $\bullet$ If $d \ne 0$, let r be the remainder when c is divided by d. Write down $(d, r)$. $\bullet$ If $d = 0$, then write down c. Once this happens, you're done, and the number you just wrote down is the greatest common divisor of a and b. 3. Repeat step 2 until you're done. For example, with $a = 7$ and $b = 4$, Euclid's algorithm computes the greatest common divisor in $4$ steps: $$(7, 4) \to (4, 3) \to (3, 1) \to (1, 0) \to 1$$ For $a > b > 0,$ compute the least value of a such that Euclid's algorithm takes $10$ steps to compute the greatest common divisor of $a$ and $b$. [b]p9.[/b] Let $ABCD$ be a square of unit side length. Inscribe a circle $C_0$ tangent to all of the sides of the square. For each positive integer $n$, draw a circle Cn that is externally tangent to $C_{n-1}$ and also tangent to sides $AB$ and $AD$. Suppose $r_i$ is the radius of circle $C_i$ for every nonnegative integer $i$. Compute $\sqrt[200]{r_0/r_{100}}$. [b]p10.[/b] Rachel and Mike are playing a game. They start at $0$ on the number line. At each positive integer on the number line, there is a carrot. At the beginning of the game, Mike picks a positive integer $n$ other than $30$. Every minute, Rachel moves to the next multiple of $30$ on the number line that has a carrot on it and eats that carrot. At the same time, every minute, Mike moves to the next multiple of $n$ on the number line that has a carrot on it and eats that carrot. Mike wants to pick an $n$ such that, as the game goes on, he is always within $1000$ units of Rachel. Compute the average (arithmetic mean) of all such $n$. [b]p11.[/b] Darryl has a six-sided die with faces $1, 2, 3, 4, 5, 6$. He knows the die is weighted so that one face comes up with probability $1/2$ and the other five faces have equal probability of coming up. He unfortunately does not know which side is weighted, but he knows each face is equally likely to be the weighted one. He rolls the die 5 times and gets a $1$, $2$, $3$, $4$ and $5$ in some unspecified order. Compute the probability that his next roll is a $6$. [b]p12.[/b] Let $F_0 = 1$, $F_1 = 1$ and $F_k = F_{k-1} + F_{k-2}$. Let $P(x) =\sum^{99}_{k=0} x^{F_k}$ . The remainder when $P(x)$ is divided by $x^3 - 1$ can be expressed as $ax^2 + bx + c$. Find $2a + b$. [b]p13.[/b] Let $\theta \ne 0$ be the smallest acute angle for which $\sin \theta$, $\sin (2\theta)$, $\sin (3\theta)$, when sorted in increasing order, form an arithmetic progression. Compute $\cos (\theta/2)$. [b]p14.[/b] A $4$-dimensional hypercube of edge length 1 is constructed in $4$-space with its edges parallel to the coordinate axes and one vertex at the origin. The coordinates of its sixteen vertices are given by $(a, b, c, d)$, where each of $a$, $b$, $c$, and $d$ is either $0$ or $1$. The $3$-dimensional hyperplane given by $x + y + z + w = 2$ intersects the hypercube at $6$ of its vertices. Compute the $3$-dimensional volume of the solid formed by the intersection. [b]p15.[/b] A student puts $2010$ red balls and $1957$ blue balls into a box. Weiqing draws randomly from the box one ball at a time without replacement. She wins if, at anytime, the total number of blue balls drawn is more than the total number of red balls drawn. Assuming Weiqing keeps drawing balls until she either wins or runs out, ompute the probability that she eventually wins. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a table $4\times 4$ we put $k$ blocks such that i) Each block covers exactly 2 cells ii) Each cell is covered by, at least, one block iii) If we delete a block; there is, at least, one cell that is not covered. Find the maximum value of $k$. Note: The blocks can overlap.