Positive integers $k, n$ and subsets $A_1, A_2, ..., A_k$ of the set $\{1, 2, ..., 2n\}$ are given. We will say that a pair of numbers $x, y$ is good, if $x<y$, $x, y\in \{1, 2, ..., 2n\}$ and there exists exactly one index $i\in \{1, 2, ..., 2n\}$, for which exactly one of $x, y$ belongs to $A_i$. Prove that there are at most $n^2$ good pairs.

Julian and Johan are playing a game with an even number of cards, say $2n$ cards, ($n \in Z_{>0}$). Every card is marked with a positive integer. The cards are shuffled and are arranged in a row, in such a way that the numbers are visible. The two players take turns picking cards. During a turn, a player can pick either the rightmost or the leftmost card. Johan is the first player to pick a card (meaning Julian will have to take the last card). Now, a player’s score is the sum of the numbers on the cards that player acquired during the game. Prove that Johan can always get a score that is at least as high as Julian’s.

In a group of $2021$ people, $1400$ of them are $\emph{saboteurs}$. Sherlock wants to find one saboteur. There are some missions that each needs exactly $3$ people to be done. A mission fails if at least one of the three participants in that mission is a saboteur! In each round, Sherlock chooses $3$ people, sends them to a mission and sees whether it fails or not. What is the minimum number of rounds he needs to accomplish his goal?

Let $n$ be a positive integer and let $(a_1,a_2,\ldots ,a_{2n})$ be a permutation of $1,2,\ldots ,2n$ such that the numbers $|a_{i+1}-a_i|$ are pairwise distinct for $i=1,\ldots ,2n-1$. Prove that $\{a_2,a_4,\ldots ,a_{2n}\}=\{1,2,\ldots ,n\}$ if and only if $a_1-a_{2n}=n$.

[u]Set 1[/u] [b]p1.[/b] Arnold is currently stationed at $(0, 0)$. He wants to buy some milk at $(3, 0)$, and also some cookies at $(0, 4)$, and then return back home at $(0, 0)$. If Arnold is very lazy and wants to minimize his walking, what is the length of the shortest path he can take? [b]p2.[/b] Dilhan selects $1$ shirt out of $3$ choices, $1$ pair of pants out of $4$ choices, and $2$ socks out of $6$ differently-colored socks. How many outfits can Dilhan select? All socks can be worn on both feet, and outfits where the only difference is that the left sock and right sock are switched are considered the same. [b]p3.[/b] What is the sum of the first $100$ odd positive integers? [b]p4.[/b] Find the sum of all the distinct prime factors of $1591$. [b]p5.[/b] Let set $S = \{1, 2, 3, 4, 5, 6\}$. From $S$, four numbers are selected, with replacement. These numbers are assembled to create a $4$-digit number. How many such $4$-digit numbers are multiples of $3$? [u]Set 2[/u] [b]p6.[/b] What is the area of a triangle with vertices at $(0, 0)$, $(7, 2)$, and $(4, 4)$? [b]p7.[/b] Call a number $n$ “warm” if $n - 1$, $n$, and $n + 1$ are all composite. Call a number $m$ “fuzzy” if $m$ may be expressed as the sum of $3$ consecutive positive integers. How many numbers less than or equal to $30$ are warm and fuzzy? [b]p8.[/b] Consider a square and hexagon of equal area. What is the square of the ratio of the side length of the hexagon to the side length of the square? [b]p9.[/b] If $x^2 + y^2 = 361$, $xy = -40$, and $x - y$ is positive, what is $x - y$? [b]p10.[/b] Each face of a cube is to be painted red, orange, yellow, green, blue, or violet, and each color must be used exactly once. Assuming rotations are indistinguishable, how many ways are there to paint the cube? [u]Set 3[/u] [b]p11.[/b] Let $D$ be the midpoint of side $BC$ of triangle $ABC$. Let $P$ be any point on segment $AD$. If $M$ is the maximum possible value of $\frac{[PAB]}{[PAC]}$ and $m$ is the minimum possible value, what is $M - m$? Note: $[PQR]$ denotes the area of triangle $PQR$. [b]p12.[/b] If the product of the positive divisors of the positive integer $n$ is $n^6$, find the sum of the $3$ smallest possible values of $n$. [b]p13.[/b] Find the product of the magnitudes of the complex roots of the equation $(x - 4)^4 +(x - 2)^4 + 14 = 0$. [b]p14.[/b] If $xy - 20x - 16y = 2016$ and $x$ and $y$ are both positive integers, what is the least possible value of $\max (x, y)$? [b]p15.[/b] A peasant is trying to escape from Chyornarus, ruled by the Tsar and his mystical faith healer. The peasant starts at $(0, 0)$ on a $6 \times 6$ unit grid, the Tsar’s palace is at $(3, 3)$, the healer is at $(2, 1)$, and the escape is at $(6, 6)$. If the peasant crosses the Tsar’s palace or the mystical faith healer, he is executed and fails to escape. The peasant’s path can only consist of moves upward and rightward along the gridlines. How many valid paths allow the peasant to escape? PS. You should use hide for answers. Rest sets have been posted [url=https://artofproblemsolving.com/community/c3h2784259p24464954]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $n$ blocks placed on the unit squares of a $n \times n$ chessboard such that there is exactly one block in each row and each column. Find the maximum value $k$, in terms of $n$, such that however the blocks are arranged, we can place $k$ rooks on the board without any two of them threatening each other. (Two rooks are not threatening each other if there is a block lying between them.)

There is a lottery at Juku’s birthday party with a number of identical prizes, where each guest can win at most one prize. It is known that if there was one prize less, then the number of possible distributions of the prizes among the guests would be $50\%$ less than it actually is, while if there was one prize more, then the number of possible distributions of the prizes would be $50\%$ more than it actually is. Find the number of possible distributions of the prizes.

Ben has $16$ balls labeled $1, 2, 3, \ldots, 16,$ as well as $4$ indistinguishable boxes. Two balls are [i]neighbors[/i] if their labels differ by $1.$ Compute the number of ways for him to put $4$ balls in each box such that each ball is in the same box as at least one of its neighbors. (The order in which the balls are placed does not matter.)

The road ministry has assigned $80$ informal companies to repair $2400$ roads. These roads connect $100$ cities to each other. Each road is between $2$ cities and there is at most $1$ road between every $2$ cities. We know that each company repairs $30$ roads that it has agencies in each $2$ ends of them. Prove that there exists a city in which $8$ companies have agencies.

Suppose there are $997$ points given in a plane. If every two points are joined by a line segment with its midpoint coloured in red, show that there are at least $1991$ red points in the plane. Can you find a special case with exactly $1991$ red points?

An open necklace can contain rubies, emeralds, and sapphires. At every step we can perform any of the following operations: [list=1] [*]We can replace two consecutive rubies with an emerald and a sapphire, where the emerald is on the left of the sapphire.[/*] [*]We can replace three consecutive emeralds with a sapphire and a ruby, where the sapphire is on the left of the ruby. [/*] [*]If we find two consecutive sapphires then we can remove them.[/*] [*]If we find consecutively and in this order a ruby, an emerald, and a sapphire, then we can remove them.[/*] [/list] Furthermore we can also reverse all of the above operations. For example by reversing 3. we can put two consecutive sapphires on any position we wish. Initially the necklace has one sapphire (and no other precious stones). Decide, with proof, whether there is a finite sequence of steps such that at the end of this sequence the necklace contains one emerald (and no other precious stones). [i]Remark:[/i] A necklace is open if its precious stones are on a line from left to right. We are not allowed to move a precious stone from the rightmost position to the leftmost as we would be able to do if the necklace was closed. [i]Proposed by Demetres Christofides, Cyprus[/i]

An $n\times n$ chessboard is given, where $n$ is an even positive integer. On every line, the unit squares are to be permuted, subject to the condition that the resulting table has to be symmetric with respect to its main diagonal (the diagonal from the top-left corner to the bottom-right corner). We say that a board is [i]alternative[/i] if it has at least one pair of complementary lines (two lines are complementary if the unit squares on them which lie on the same column have distinct colours). Otherwise, we call the board [i]nonalternative[/i]. For what values of $n$ do we always get from the $n\times n$ chessboard an alternative board?\\ \\ [i](Alexandru Petrescu and Andra Elena Mircea)[/i]

Georg has a circular game board with 100 squares labelled $1, 2, . . . , 100$. Georg chooses three numbers $a, b, c$ among the numbers $1, 2, . . . , 99$. The numbers need not be distinct. Initially there is a piece on the square labelled $100$. First, Georg moves the piece $a$ squares forward $33$ times and puts a caramel on each of the squares the piece lands on. Then he moves the piece $b$ squares forward $33$ times and puts a caramel on each of the squares the piece lands on. Finally, he moves the piece $c$ squares forward $33$ times and puts a caramel on each of the squares the piece lands on. Thus he puts a total of $99$ caramels on the board. Georg wins all the caramels on square number $1$. How many caramels can Georg win, at most? [img]https://cdn.artofproblemsolving.com/attachments/d/c/af438e5feadca5b1bfc98ae427f6fc24655e29.png[/img]

There are 100 persons in a hall, everyone knowing at least 66 of the others. Prove that there is a case in which among any four some two don’t know each other.

$100$ heaps of stones lie on a table. Two players make moves in turn. At each move, a player can remove any non-zero number of stones from the table, so that at least one heap is left untouched. The player that cannot move loses. Determine, for each initial position, which of the players, the first or the second, has a winning strategy. [i]K. Kokhas[/i] [b]EDIT.[/b] It is indeed confirmed by the sender that empty heaps are still heaps, so the third post contains the right guess of an interpretation.

Ten localities are served by two international airlines such that there exists a direct service (without stops) between any two of these localities and all airline schedules offer round-trip service between the cities they serve. Prove that at least one of the airlines can offer two disjoint round trips each containing an odd number of landings.

We call a diagonal of a polygon [i]nice[/i], if it is entirely inside the polygon or entirely outside the polygon. Let $P$ be an $n$–gon with no three of its vertices being on the same line. Prove that $P$ has at least $3(n-3)/2$ nice diagonals. [i]Proposed by Bálint Hujter, Budapest and Gábor Szűcs, Szikszó[/i]

Given a board consists of $n \times n$ unit squares ($n \ge 3$). Each unit square is colored black and white, resembling a chessboard. In each step, TOMI can choose any $2 \times 2$ square and change the color of every unit square chosen with the other color (white becomes black and black becomes white). Find every $n$ such that after a finite number of moves, every unit square on the board has a same color.

[b]p1.[/b] Warren interrogates the $25$ members of his cabinet, each of whom always lies or always tells the truth. He asks them all, “How many of you always lie?” He receives every integer answer from $1$ to $25$ exactly once. Find the actual number of liars in his cabinet. [b]p2.[/b] Abraham thinks of distinct nonzero digits $E$, $M$, and $C$ such that $E +M = \overline{CC}$. Help him evaluate the sum of the two digit numbers $\overline{EC}$ and $\overline{MC}$. (Note that $\overline{CC}$, $\overline{EC}$, and $\overline{MC}$ are read as two-digit numbers.) [b]p3.[/b] Let $\omega$, $\Omega$, $\Gamma$ be concentric circles such that $\Gamma$ is inside $\Omega$ and $\Omega$ is inside $\omega$. Points $A,B,C$ on $\omega$ and $D,E$ on $\Omega$ are chosen such that line $AB$ is tangent to $\Omega$, line $AC$ is tangent to $\Gamma$, and line $DE$ is tangent to $\Gamma$. If $AB = 21$ and $AC = 29$, find $DE$. [b]p4.[/b] Let $a$, $b$, and $c$ be three prime numbers such that $a + b = c$. If the average of two of the three primes is four less than four times the fourth power of the last, find the second-largest of the three primes. [b]p5.[/b] At Stillwells Ice Cream, customers must choose one type of scoop and two different types of toppings. There are currently $630$ different combinations a customer could order. If another topping is added to the menu, there would be $840$ different combinations. If, instead, another type of scoop were added to the menu, compute the number of different combinations there would be. [b]p6.[/b] Eleanor the ant takes a path from $(0, 0)$ to $(20, 24)$, traveling either one unit right or one unit up each second. She records every lattice point she passes through, including the starting and ending point. If the sum of all the $x$-coordinates she records is $271$, compute the sum of all the $y$-coordinates. (A lattice point is a point with integer coordinates.) [b]p7.[/b] Teddy owns a square patch of desert. He builds a dam in a straight line across the square, splitting the square into two trapezoids. The perimeters of the trapezoids are$ 64$ miles and $76$ miles, and their areas differ by $135$ square miles. Find, in miles, the length of the segment that divides them. [b]p8.[/b] Michelle is playing Spot-It with a magical deck of $10$ cards. Each card has $10$ distinct symbols on it, and every pair of cards shares exactly $1$ symbol. Find the minimum number of distinct symbols on all of the cards in total. [b]p9.[/b] Define the function $f(n) = \frac{1}{2^n} + \frac{1}{3^n} + \frac{1}{4^n} + ...$ for integers $n \ge 2$. Find $$f(2) + f(4) + f(6) + ... .$$ [b]p10.[/b] There are $9$ indistinguishable ants standing on a $3\times 3$ square grid. Each ant is standing on exactly one square. Compute the number of different ways the ants can stand so that no column or row contains more than $3$ ants. [b]p11.[/b] Let $s(N)$ denote the sum of the digits of $N$. Compute the sum of all two-digit positive integers $N$ for which $s(N^2) = s(N)^2$. [b]p12.[/b] Martha has two square sheets of paper, $A$ and $B$. With each sheet, she repeats the following process four times: fold bottom side to top side, fold right side to left side. With sheet $A$, she then makes a cut from the top left corner to the bottom right. With sheet $B$, she makes a cut from the bottom left corner to the top right. Find the total number of pieces of paper yielded from sheets $A$ and sheets $B$. [img]https://cdn.artofproblemsolving.com/attachments/f/6/ff3a459a135562002aa2c95067f3f01441d626.png[/img] [b]p13.[/b] Let $x$ and $y$ be positive integers such that gcd $(x^y, y^x) = 2^{28}$. Find the sum of all possible values of min $(x, y)$. [b]p14.[/b] Convex hexagon $TRUMAN$ has opposite sides parallel. If each side has length $3$ and the area of this hexagon is $5$, compute $$TU \cdot RM \cdot UA \cdot MN \cdot AT \cdot NR.$$ [b]p15.[/b] Let $x$, $y$, and $z$ be positive real numbers satisfying the system $$\begin{cases} x^2 + xy + y^2 = 25\\ y^2 + yz + z^2 = 36 \\ z^2 + zx + x^2 = 49 \end{cases}$$ Compute $x^2 + y^2 + z^2$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a $2025 \times 2025$ table, several cells are marked. At each move, Cyril can get to know the number of marked cells in any checkered square inside the initial table, with side less than $2025$. What is the minimal number of moves, which allows to determine the total number of marked cells for sure? (5 marks)

Given $2n$ natural numbers, so that the average arithmetic of those $2n$ number is $2$. If all the number is not more than $2n$. Prove we can divide those $2n$ numbers into $2$ sets, so that the sum of each set to be the same.

In a ballroom dance class $15$ boys and $15$ girls are lined up in parallel rows so that $15$ couples are formed. It so happens that the difference in height between the boy and the girl in each couple is not more than $10$ cm. Prove that if the boys and the girls were placed in each line in order of decreasing height, then the difference in height in each of the newly formed couples would still be at most $10$ cm. (AG Pechkovskiy, Moscow)

In the land of Cockaigne, people play the following solitaire. It starts from a finite string of zeros and ones, and are granted the following moves: (i) cancel each two consecutive ones; (ii) delete three consecutive zeros; (iii) if the substring within the string is $01$, one may replace this by substring $100$. The moves (i), (ii) and (iii) must be made one at a time. You win if you can reduce the string to a string formed by two digits or less. (For example, starting from $0101$, one can win using move (iii) first in the last two digits, resulting in $01100$, then playing the move (i) on two 'ones', and finally the move (ii) on the three zeros, one will get the empty string.) Among all the $1024$ possible strings of ten-digit binary numbers, how many are there from which it is not possible to win the solitary?

A single segment contains several non-intersecting red segments, the total length of which is greater than $0.5$. Are there necessarily two red dots at the distance: a) $1/99$ b) $1/100$ ?

Find the number of ways that a positive integer $n$ can be represented as a sum of one or more consecutive positive integers.