For even positive integer $n$ we put all numbers $1,2,...,n^2$ into the squares of an $n\times n$ chessboard (each number appears once and only once). Let $S_1$ be the sum of the numbers put in the black squares and $S_2$ be the sum of the numbers put in the white squares. Find all $n$ such that we can achieve $\frac{S_1}{S_2}=\frac{39}{64}.$

It is given a graph whose vertices are positive integers and an edge between numbers $a$ and $b$ exists if and only if $a + b + 1 | a^2 + b^2 + 1$. Is this graph connected?

There are $k$ piles of stones with $2020$ stones in each pile. Amber can choose any two non-empty piles of stones, and Barbara can take one stone from one of the two chosen piles and puts it into the other pile. Amber wins if she can eventually make an empty pile. What is the least $k$ such that Amber can always win?

What is the largest number of regions into which a plane can be divided by drawing $n$ pairs of parallel lines?

Let $A_{1}A_{2}B_{1}B_{2}$ be a convex quadrilateral. At adjacent vertices $A_{1}$ and $A_{2}$ there are two Argentinian cities. At adjacent vertices $B_{1}$ and $B_{2}$ there are two Brazilian cities. There are $a$ Argentinian cities and $b$ Brazilian cities in the quadrilateral interior, no three of which collinear. Determine if it's possible, independently from the cities position, to build straight roads, each of which connects two Argentinian cities ou two Brazilian cities, such that: $\bullet$ Two roads does not intersect in a point which is not a city; $\bullet$ It's possible to reach any Argentinian city from any Argentinian city using the roads; and $\bullet$ It's possible to reach any Brazilian city from any Brazilian city using the roads. If it's always possible, construct an algorithm that builds a possible set of roads.

Let be given an $n \times n$ chessboard, $n \in N$. We wish to tile it using particular tetraminos which can be rotated. For which $n$ is this possible if we use (a) $T$-tetraminos (b) both kinds of $L$-tetraminos?

A connected graph with at least one vertex of an odd degree is given. Show that one can color the edges of the graph red and blue in such a way that, for each vertex, the absolute difference between the numbers of red and blue edges at that vertex does not exceed 1.

Baron and Peter are playing a game. They are given a simple finite graph $G$ with $n\ge 3$ vertex and $k$ edges that connects the vertices. First Peter labels two vertices A and B, and places a counter at A. Baron starts first. A move for Baron is move the counter along an edge. Peter's move is to remove an edge from the graph. Baron wins if he reaches $B$, otherwise Peter wins. Given the value of $n$, what is the largest $k$ so that Peter can always win?

[u]Round 5[/u] [b]p13.[/b] You have a $26 \times 26$ grid of squares. Color each randomly with red, yellow, or blue. What is the expected number (to the nearest integer) of $2 \times 2$ squares that are entirely red? [b]p14.[/b] Four snakes are boarding a plane with four seats. Each snake has been assigned to a different seat. The first snake sits in the wrong seat. Any subsequent snake will sit in their assigned seat if vacant, if not, they will choose a random seat that is available. What is the expected number of snakes who sit in their correct seats? [b]p15.[/b] Let $n \ge 1$ be an integer and $a > 0$ a real number. In terms of n, find the number of solutions $(x_1, ..., x_n)$ of the equation $\sum^n_{i=1}(x^2_i + (a - x_i)^2) = na^2$ such that $x_i$ belongs to the interval $[0, a]$ , for $i = 1, 2, . . . , n$. [u]Round 6 [/u] [b]p16.[/b] All roots of $$\prod^{25}_{n=1} \prod^{2n}_{k=0}(-1)^k \cdot x^k = 0$$ are written in the form $r(\cos \phi + i\sin \phi)$ for $i^2 = -1$, $r > 0$, and $0 \le \phi < 2\pi$. What is the smallest positive value of $\phi$ in radians? [b]p17.[/b] Find the sum of the distinct real roots of the equation $$\sqrt[3]{x^2 - 2x + 1} + \sqrt[3]{x^2 - x - 6} = \sqrt[3]{2x^2 - 3x - 5}.$$ [b]p18.[/b] If $a$ and $b$ satisfy the property that $a2^n + b$ is a square for all positive integers $n$, find all possible value(s) of $a$. [u]Round 7 [/u] [b]p19.[/b] Compute $(1 - \cot 19^o)(1 - \cot 26^o)$. [b]p20.[/b] Consider triangle $ABC$ with $AB = 3$, $BC = 5$, and $\angle ABC = 120^o$. Let point $E$ be any point inside $ABC$. The minimum of the sum of the squares of the distances from $E$ to the three sides of $ABC$ can be written in the form $a/b$ , where a and b are natural numbers such that the greatest common divisor of $a$ and $b$ is $1$. Find $a + b$. [b]p21.[/b] Let $m \ne 1$ be a square-free number (an integer – possibly negative – such that no square divides $m$). We denote $Q(\sqrt{m})$ to be the set of all $a + b\sqrt{m}$ where $a$ and $b$ are rational numbers. Now for a fixed $m$, let $S$ be the set of all numbers $x$ in $Q(\sqrt{m})$ such that x is a solution to a polynomial of the form: $x^n + a_1x^{n-1} + .... + a_n = 0$, where $a_0$, $...$, $a_n$ are integers. For many integers m, $S = Z[\frac{m}] = \{a + b\sqrt{m}\}$ where $a$ and $b$ are integers. Give a classification of the integers for which this is not true. (Hint: It is true for $ m = -1$ and $2$.) PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2782002p24434611]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

a) A positive integer is written at each vertex of a triangle. Then on each side of the triangle the greatest common divisor of its ends is written. It is possible that the numbers written on the sides be three consecutive integers, in some order? b) A positive integer is written at each vertex of a tetrahedron. Then, on each edge of the tetrahedron is written the greatest common divisor of its ends . It is possible that the numbers written in the edges are six consecutive integers, in some order?

There are $16$ competitors in a tournament, all of whom have different playing strengths and in any match between two players the stronger player always wins. Show that it is possible to find the strongest and second strongest players in $18$ matches.

There was a rook at some square of a $10 \times 10{}$ chessboard. At each turn it moved to a square adjacent by side. It visited each square exactly once. Prove that for each main diagonal (the diagonal between the corners of the board) the following statement is true: in the rook’s path there were two consecutive steps at which the rook first stepped away from the diagonal and then returned back to the diagonal. [i]Alexandr Gribalko[/i]

The circumference of a circle is divided into $45$ arcs, each of length $1.$ Initially, there are $15$ snakes, each of length $1,$ occupying every third arc. Every second, each snake independently moves either one arc left or one arc right, each with probability $\tfrac{1}{2}.$ If two snakes ever touch, they merge to form a single snake occupying the arcs of both of the previous snakes, and the merged snake moves as one snake. Compute the expected number of seconds until there is only one snake left.

Some of the points in the plane are white and some are blue (every point of the plane is either white or blue). Prove that for every positive number $r$: (a) there are at least two points with different color such that the distance between them is equal to $r$; (b) there are at least two points with the same color and the distance between them is equal to $r$; (c) will the statements above be true if the plane is replaced with the real line?

[i]25 problems for 30 minutes[/i] [b]p1.[/b] You and nine friends spend $4000$ dollars on tickets to attend the new Harry Styles concert. Unfortunately, six friends cancel last minute due to the u. You and your remaining friends still attend the concert and split the original cost of $4000$ dollars equally. What percent of the total cost does each remaining individual have to pay? [b]p2.[/b] Find the number distinct $4$ digit numbers that can be formed by arranging the digits of $2021$. [b]p3.[/b] On a plane, Darnay draws a triangle and a rectangle such that each side of the triangle intersects each side of the rectangle at no more than one point. What is the largest possible number of points of intersection of the two shapes? [b]p4.[/b] Joy is thinking of a two-digit number. Her hint is that her number is the sum of two $2$-digit perfect squares $x_1$ and $x_2$ such that exactly one of $x_i - 1$ and $x_i + 1$ is prime for each $i = 1, 2$. What is Joy's number? [b]p5.[/b] At the North Pole, ice tends to grow in parallelogram structures of area $60$. On the other hand, at the South Pole, ice grows in right triangular structures, in which each triangular and parallelogram structure have the same area. If every ice triangle $ABC$ has legs $\overline{AB}$ and $\overline{AC}$ that are integer lengths, how many distinct possible lengths are there for the hypotenuse $\overline{BC}$? [b]p6.[/b] Carlsen has some squares and equilateral triangles, all of side length $1$. When he adds up the interior angles of all shapes, he gets $1800^o$. When he adds up the perimeters of all shapes, he gets $24$. How many squares does he have? [b]p7.[/b] Vijay wants to hide his gold bars by melting and mixing them into a water bottle. He adds $100$ grams of liquid gold to $100$ grams of water. His liquefied gold bars have a density of $20$ g/ml and water has a density of $1$ g/ml. Given that the density of the mixture in g/mL can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute the sum $m + n$. (Note: density is mass divided by volume, gram (g) is unit of mass and ml is unit of volume. Further, assume the volume of the mixture is the sum of the volumes of the components.) [b]p8.[/b] Julius Caesar has epilepsy. Specifically, if he sees $3$ or more flashes of light within a $0.1$ second time frame, he will have a seizure. His enemy Brutus has imprisoned him in a room with $4$ screens, which flash exactly every $4$, $5$, $6$, and $7$ seconds, respectively. The screens all flash at once, and $105$ seconds later, Caesar opens his eyes. How many seconds after he opened his eyes will Caesar first get a seizure? [b]p9.[/b] Angela has a large collection of glass statues. One day, she was bored and decided to use some of her statues to create an entirely new one. She melted a sphere with radius $12$ and a cone with height of 18 and base radius of $2$. If Angela wishes to create a new cone with a base radius $2$, what would the the height of the newly created cone be? [b]p10.[/b] Find the smallest positive integer $N$ satisfying these properties: (a) No perfect square besides $1$ divides $N$. (b) $N$ has exactly $16$ positive integer factors. [b]p11.[/b] The probability of a basketball player making a free throw is $\frac15$. The probability that she gets exactly $2$ out of $4$ free throws in her next game can be expressed as $\frac{m}{n}$ for relatively prime positive integers m and n. Find $m + n$. [b]p12.[/b] A new donut shop has $1000$ boxes of donuts and $1000$ customers arriving. The boxes are numbered $1$ to $1000$. Initially, all boxes are lined up by increasing numbering and closed. On the first day of opening, the first customer enters the shop and opens all the boxes for taste testing. On the second day of opening, the second customer enters and closes every box with an even number. The third customer then "reverses" (if closed, they open it and if open, they close it) every box numbered with a multiple of three, and so on, until all $1000$ customers get kicked out for having entered the shop and reversing their set of boxes. What is the number on the sixth box that is left open? [b]p13.[/b] For an assignment in his math class, Michael must stare at an analog clock for a period of $7$ hours. He must record the times at which the minute hand and hour hand form an angle of exactly $90^o$, and he will receive $1$ point for every time he records correctly. What is the maximum number of points Michael can earn on his assignment? [b]p14.[/b] The graphs of $y = x^3 +5x^2 +4x-3$ and $y = -\frac15 x+1$ intersect at three points in the Cartesian plane. Find the sum of the $y$-coordinates of these three points. [b]p15.[/b] In the quarterfinals of a single elimination countdown competition, the $8$ competitors are all of equal skill. When any $2$ of them compete, there is exactly a $50\%$ chance of either one winning. If the initial bracket is randomized, the probability that two of the competitors, Daniel and Anish, face off in one of the rounds can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p$, $q$. Find $p + q$. [b]p16.[/b] How many positive integers less than or equal to $1000$ are not divisible by any of the numbers $2$, $3$, $5$ and $11$? [b]p17.[/b] A strictly increasing geometric sequence of positive integers $a_1, a_2, a_3,...$ satisfies the following properties: (a) Each term leaves a common remainder when divided by $7$ (b) The first term is an integer from $1$ to $6$ (c) The common ratio is an perfect square Let $N$ be the smallest possible value of $\frac{a_{2021}}{a_1}$. Find the remainder when $N$ is divided by $100$. [b]p18.[/b] Suppose $p(x) = x^3 - 11x^2 + 36x - 36$ has roots $r, s,t$. Find %\frac{r^2 + s^2}{t}+\frac{s^2 + t^2}{r}+\frac{t^2 + r^2}{s}%. [b]p19.[/b] Let $a, b \le 2021$ be positive integers. Given that $ab^2$ and $a^2b$ are both perfect squares, let $G = gcd(a, b)$. Find the sum of all possible values of $G$. [b]p20.[/b] Jessica rolls six fair standard six-sided dice at the same time. Given that she rolled at least four $2$'s and exactly one $3$, the probability that all six dice display prime numbers can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$, $n$. What is $m + n$? [b]p21.[/b] Let $a, b, c$ be numbers such $a + b + c$ is real and the following equations hold: $$a^3 + b^3 + c^3 = 25$$ $$\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}= 1$$ $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{25}{9}$$ The value of $a + b + c$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$, $n$. Find $m + n$. [b]p22.[/b] Let $\omega$ be a circle and $P$ be a point outside $\omega$. Let line $\ell$ pass through $P$ and intersect $\omega$ at points $A,B$ and with $PA < PB$ and let $m$ be another line passing through $P$ intersecting $\omega$ at points $C,D$ with $PC < PD$. Let X be the intersection of $AD$ and $BC$. Given that $\frac{PC}{CD}=\frac23$, $\frac{PC}{PA}=\frac45$, and $\frac{[ABC]}{[ACD]}=\frac79$,the value of $\frac{[BXD]}{[BXA]}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$: Find $m + n$. [b]p23.[/b] Define the operation $a \circ b =\frac{a^2 + 2ab + a - 12}{b}$. Given that $1 \circ (2 \circ (3 \circ (... 2019 \circ (2020 \circ 2021)))...)$ can be expressed as $-\frac{a}{b}$ for some relatively prime positive integers $a,b$, compute $a + b$. [b]p24.[/b] Find the largest integer $n \le 2021$ for which $5^{n-3} | (n!)^4$ [b]p25.[/b] On the Cartesian plane, a line $\ell$ intersects a parabola with a vertical axis of symmetry at $(0, 5)$ and $(4, 4)$. The focus $F$ of the parabola lies below $\ell$, and the distance from $F$ to $\ell$ is $\frac{16}{\sqrt{17}}$. Let the vertex of the parabola be $(x, y)$. The sum of all possible values of $y$ can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p, q$. Find $p + q$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] What is the value of the sum $2 + 20 + 202 + 2022$? [b]p2.[/b] Find the smallest integer greater than $10000$ that is divisible by $12$. [b]p3.[/b] Valencia chooses a positive integer factor of $6^{10}$ at random. The probability that it is odd can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime integers. Find $m + n$. [b]p4.[/b] How many three digit positive integers are multiples of $4$ but not $8$? [b]p5.[/b] At the Jane Street store, Andy accidentally buys $5$ dollars more worth of shirts than he had planned. Originally, including the tip to the cashier, he planned to spend all of the remaining $90$ dollars on his giftcard. To compensate for his gluttony, Andy instead gives the cashier a smaller, $12.5\%$ tip so that he still spends $90$ dollars total. How much percent tip was Andy originally planning on giving? [b]p6.[/b] Let $A,B,C,D$ be four coplanar points satisfying the conditions $AB = 16$, $AC = BC =10$, and $AD = BD = 17$. What is the minimum possible area of quadrilateral $ADBC$? [b]p7.[/b] How many ways are there to select a set of three distinct points from the vertices of a regular hexagon so that the triangle they form has its smallest angle(s) equal to $30^o$? [b]p8.[/b] Jaeyong rolls five fair $6$-sided die. The probability that the sum of some three rolls is exactly $8$ times the sum of the other two rolls can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]p9.[/b] Find the least positive integer n for there exists some positive integer $k > 1$ for which $k$ and $k + 2$ both divide $\underbrace{11...1}_{n\,\,\,1's}$. [b]p10.[/b] For some real constant $k$, line $y = k$ intersects the curve $y = |x^4-1|$ four times: points $A$,$B$,$C$ and $D$, labeled from left to right. If $BC = 2AB = 2CD$, then the value of $k$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]p11.[/b] Let a be a positive real number and $P(x) = x^2 -8x+a$ and $Q(x) = x^2 -8x+a+1$ be quadratics with real roots such that the positive difference of the roots of $P(x)$ is exactly one more than the positive difference of the roots of $Q(x)$. The value of a can be written as a common fraction $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [b]p12.[/b] Let $ABCD$ be a trapezoid satisfying $AB \parallel CD$, $AB = 3$, $CD = 4$, with area $35$. Given $AC$ and $BD$ intersect at $E$, and $M$, $N$, $P$, $Q$ are the midpoints of segments $AE$,$BE$,$CE$,$DE$, respectively, the area of the intersection of quadrilaterals $ABPQ$ and $CDMN$ can be expressed as $\frac{m}{n}$ where $m, n$ are relatively prime positive integers. Find $m + n$. [b]p13.[/b] There are $8$ distinct points $P_1, P_2, ... , P_8$ on a circle. How many ways are there to choose a set of three distinct chords such that every chord has to touch at least one other chord, and if any two chosen chords touch, they must touch at a shared endpoint? [b]p14.[/b] For every positive integer $k$, let $f(k) > 1$ be defined as the smallest positive integer for which $f(k)$ and $f(k)^2$ leave the same remainder when divided by $k$. The minimum possible value of $\frac{1}{x}f(x)$ across all positive integers $x \le 1000$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $m + n$. [b]p15.[/b] In triangle $ABC$, let $I$ be the incenter and $O$ be the circumcenter. If $AO$ bisects $\angle IAC$, $AB + AC = 21$, and $BC = 7$, then the length of segment $AI$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Every cell of a $20 \times 20$ table has to be coloured black or white (there are $2^{400}$ such colourings in total). Given any colouring $P$, we consider division of the table into rectangles with sides in the grid lines where no rectangle contains more than two black cells and where the number of rectangles containing at most one black cell is the least possible. We denote this smallest possible number of rectangles containing at most one black cell by $f(P)$. Determine the maximum value of $f(P)$ as $P$ ranges over all colourings.

The integer number $n > 1$ is given and a set $S \subset \{0, 1, 2, \ldots, n-1\}$ with $|S| > \frac{3}{4} n$. Prove that there exist integer numbers $a, b, c$ such that the remainders after the division by $n$ of the numbers: \[a, b, c, a+b, b+c, c+a, a+b+c\] belong to $S$.

Alphonse and Beryl play a game starting with a blank blackboard. Alphonse goes first and the two players alternate turns. On Alphonse's first turn, he writes the integer $10^{2011}$ on the blackboard. On each subsequent turn, each player can do exactly one of the following two things: [b](i)[/b] replace any number $x$ that is currently on the blackboard with two integers a and b greater than $1$ such that $x = ab,$ or [b](ii)[/b] erase one or two copies of a number $y$ that appears at least twice on the blackboard. Thus, there may be many numbers on the board at any time. The first player who cannot do either of these things loses. Determine which player has a winning strategy and explain the strategy.

Set $T\subset\{1,2,\dots,n\}^3$ has the property that for any two triplets $(a,b,c)$ and $(x,y,z)$ in $T$, we have $a<b<c$, and also, we know that at most one of the equalities $a=x$, $b=y$, $c=z$ holds. Maximize $|T|$.

Find the formula to express the number of $ n\minus{}$series of letters which contain an even number of vocals (A,I,U,E,O).

Let $n$ be a positive integer, and let $S_1,S_2,…,S_n$ be a collection of finite non-empty sets such that $$\sum_{1\leq i<j\leq n}{\frac{|S_i \cap S_j|}{|S_i||S_j|}} <1.$$ Prove that there exist pairwise distinct elements $x_1,x_2,…,x_n$ such that $x_i$ is a member of $S_i$ for each index $i$.

50 knights of King Arthur sat at the Round Table. A glass of white or red wine stood before each of them. It is known that at least one glass of red wine and at least one glass of white wine stood on the table. The king clapped his hands twice. After the first clap every knight with a glass of red wine before him took a glass from his left neighbour. After the second clap every knight with a glass of white wine (and possibly something more) before him gave this glass to the left neughbour of his left neighbour. Prove that some knight was left without wine. [i]Proposed by A. Khrabrov, incorrect translation from Hungarian[/i]

On fields of $n \times n$ chessboard $n^2$ different integers have been arranged, one in each field. In each column, field with biggest number was colored in red. Set of $n$ fields of chessboard name [i]admissible[/i], if no two of that fields aren't in the same row and aren't in the same column. From all admissible sets, set with biggest sum of numbers in it's fields has been chosen. Prove that red field is in this set.

$2n$ students take part in a math competition. First, each of the students sends its task to the members of the jury, after which each of the students receives from the jury one of proposed tasks (all received tasks are different). Let's call the competition [i]honest[/i], if there are $n$ students who were given the tasks suggested by the remaining $n$ participants. Prove that the number of task distributions in which the competition is honest is a square of natural numbers.