A collection of cardboard circles, each with a diameter of at most $1$, lie on a $5\times 8$ table without overlapping or overhanging the edge of the table. A cardboard circle of diameter $2$ is added to the collection. Prove that this new collection of cardboard circles can be placed on a $7\times 7$ table without overlapping or overhanging the edge.

There were $2n,$ $n\ge2,$ teams in a tournament. Each team played against every other team once without draws. A team gets 0 points for a loss and gets as many points for a win as its current number of losses. For which $n$ all the teams could end up with the same non-zero number of points?

The Tower of Hanoi is a puzzle with $n$ disks of different sizes and $3$ vertical rods on it. All of the disks are initially placed on the leftmost rod, sorted by size such that the largest disk is on the bottom. On each turn, one may move the topmost disk of any nonempty rod onto any other rod, provided that it is smaller than the current topmost disk of that rod, if it exists. (For instance, if there were two disks on different rods, the smaller disk could move to either of the other two rods, but the larger disk could only move to the empty rod.) The puzzle is solved when all of the disks are moved to the rightmost rod. The specifications normally include an intelligent monk to move the disks, but instead there is a monkey making random moves (with each valid move having an equal probability of being selected). Given $64$ disks, what is the expected number of moves the monkey will have to make to solve the puzzle?

[b]p1.[/b] For $25$ bagels they paid as many rubles as the number of bagels you can buy with a ruble. How much does one bagel cost? [b]p2.[/b] Cut the square into the figure into$ 4$ parts of the same shape and size so that each part contains exactly one shaded square. [img]https://cdn.artofproblemsolving.com/attachments/a/2/14f0d435b063bcbc55d3dbdb0a24545af1defb.png[/img] [b]p3.[/b] The numerator and denominator of the fraction are positive numbers. The numerator is increased by $1$, and the denominator is increased by $10$. Can this increase the fraction? [b]p4.[/b] The brother left the house $5$ minutes later than his sister, following her, but walked one and a half times faster than her. How many minutes after leaving will the brother catch up with his sister? [b]p5.[/b] Three apples are worth more than five pears. Can five apples be cheaper than seven pears? Can seven apples be cheaper than thirteen pears? (All apples cost the same, all pears too.) [b]p6.[/b] Give an example of a natural number divisible by $6$ and having exactly $15$ different natural divisors (counting $1$ and the number itself). [b]p7.[/b] In a round dance, $30$ children stand in a circle. Every girl's right neighbor is a boy. Half of the boys have a boy on their right, and all the other boys have a girl on their right. How many boys and girls are there in a round dance? [b]p8.[/b] A sheet of paper was bent in half in a straight line and pierced with a needle in two places, and then unfolded and got $4$ holes. The positions of three of them are marked in figure Where might the fourth hole be? [img]https://cdn.artofproblemsolving.com/attachments/c/8/53b14ddbac4d588827291b27c40e3f59eabc24.png[/img] [b]p9 [/b] The numbers 1$, 2, 3, 4, 5, _, 2000$ are written in a row. First, third, fifth, etc. crossed out in order. Of the remaining $1000 $ numbers, the first, third, fifth, etc. are again crossed out. They do this until one number remains. What is this number? [b]p10.[/b] On the number axis there lives a grasshopper who can jump $1$ and $4$ to the right and left. Can he get from point $1$ to point $2$ of the numerical axis in $1996$ jumps if he must not get to points with coordinates divisible by $4$ (points $0$, $\pm 4$, $\pm 8$ etc.)? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

$120$ pirates distribute $119$ gold pieces among themselves. Then the captain checks if any pirate has $15$ or more gold pieces. If he finds the first one, he must give all his gold pieces to other pirates, whereby he may not give more than one gold piece to anyone. This control is repeated as long as there is any pirate with $15$ or more gold pieces. Does this process end after a lot of checks?

For positive integer $a,b,c,d$ there are $a+b+c+d$ points on plane which none of three are collinear. Prove there exist two lines $l_1, l_2 $ such that (1) $l_1, l_2 $ are not parallel. (2) $l_1, l_2 $ do not pass through any of $a+b+c+d$ points. (3) There are $ a, b, c, d $ points on each region separated by two lines $l_1, l_2 $.

Let $n$ be some natural number. One boss writes $n$ letters a day numerated from 1 to $n$ consecutively. When he writes a letter he piles it up (on top) in a box. When his secretary is free, she gets the letter on the top of the pile and prints it. Sometimes the secretary isn’t able to print the letter before her boss puts another one or more on the pile in the box. Though she is always able to print all of the letters at the end of the day. A permutation is called [i]“printable”[/i] if it is possible for the letters to be printed in this order. Find a formula for the number of [i]“printable”[/i] permutations.

Find all positive integer $n\ge 3$ such that it is possible to mark the vertices of a regular $n$- gon with the number from 1 to n so that for any three vertices $A, B$ and $C$ with $AB = AC$, the number in $A$ is greater or smaller than both numbers in $B, C$.

Nine lattice points (i.e. with integer coordinates) $P_1,P_2,...,P_9$ are given in space. Show that the midpoint of at least one of the segments $P_iP_j$ , where $1 \le i < j \le 9$, is a lattice point as well.

Let $n\geq 3$ be a positive integer. Consider an $n\times n$ square. In each cell of the square, one of the numbers from the set $M=\{1,2,\ldots,2n-1\}$ is to be written. One such filling is called [i]good[/i] if, for every index $1\leq i\leq n,$ row no. $i$ and column no. $i,$ together, contain all the elements of $M$. [list=a] [*]Prove that there exists $n\geq 3$ for which a good filling exists. [*]Prove that for $n=2017$ there is no good filling of the $n\times n$ square. [/list]

Suppose $S=\{1,2,3,...,2017\}$,for every subset $A$ of $S$,define a real number $f(A)\geq 0$ such that: $(1)$ For any $A,B\subset S$,$f(A\cup B)+f(A\cap B)\leq f(A)+f(B)$; $(2)$ For any $A\subset B\subset S$, $f(A)\leq f(B)$; $(3)$ For any $k,j\in S$,$$f(\{1,2,\ldots,k+1\})\geq f(\{1,2,\ldots,k\}\cup \{j\});$$ $(4)$ For the empty set $\varnothing$, $f(\varnothing)=0$. Confirm that for any three-element subset $T$ of $S$,the inequality $$f(T)\leq \frac{27}{19}f(\{1,2,3\})$$ holds.

A game is played on an $m \times n$ chessboard. At the beginning, there is a coin on one of the squares. Two players take turns to move the coin to an adjacent square (horizontally or vertically). The coin may never be moved to a square that has been occupied before. If a player cannot move any more, he loses. Prove: [list] [*] If the size (number of squares) of the board is even, then the player to move first has a winning strategy, regardless of the initial position. [*] If the size of the board is odd, then the player to move first has a winning strategy if and only if the coin is initially placed on a square whose colour is not the same as the colour of the corners. [/list]

$k$ vertices of a regular $n$-gon $P$ are coloured. A colouring is called almost uniform if for every positive integer $m$ the following condition is satisfied: If $M_1$ is a set of m consecutive vertices of $P$ and $M_2$ is another such set then the number of coloured vertices of $M_1$ differs from the number of coloured vertices of $M_2$ at most by $1$. Prove that for all positive integers $k$ and $n$ ($k \le n$) an almost uniform colouring exists and that it is unique within a rotation. (M Kontsevich, Moscow)

A square is triangulated in such way that no three vertices are collinear. For every vertex (including vertices of the square) the number of sides issuing from it is counted. Can it happen that all these numbers are even?

Three gamblers play against each other for money. They each start by placing a pile of one-krone coins on the table, and from this point on the total number of coins on the table does not change. The ratio between the number of coins they start with is $6 : 5 : 4$. At the end of the game, the ratio of the number of coins they have is $7 : 6 : 5$ in some order. At the end of the game, one of the gamblers has three coins more than at the beginning. How many coins does this gambler have at the end?

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Compute $(2 + 0)(2 + 2)(2 + 0)(2 + 2)$. [b]p2.[/b] Given that $25\%$ of $x$ is $120\%$ of $30\%$ of $200$, find $x$. [b]p3.[/b] Jacob had taken a nap. Given that he fell asleep at $4:30$ PM and woke up at $6:23$ PM later that same day, for how many minutes was he asleep? [b]p4.[/b] Kevin is painting a cardboard cube with side length $12$ meters. Given that he needs exactly one can of paint to cover the surface of a rectangular prism that is $2$ meters long, $3$ meters wide, and $6$ meters tall, how many cans of paint does he need to paint the surface of his cube? [b]p5.[/b] How many nonzero digits does $200 \times 25 \times 8 \times 125 \times 3$ have? [b]p6.[/b] Given two real numbers $x$ and $y$, define $x \# y = xy + 7x - y$. Compute the absolute value of $0 \# (1 \# (2 \# (3 \# 4)))$. [b]p7.[/b] A $3$-by-$5$ rectangle is partitioned into several squares of integer side length. What is the fewest number of such squares? Squares in this partition must not overlap and must be contained within the rectangle. [b]p8.[/b] Points $A$ and $B$ lie in the plane so that $AB = 24$. Given that $C$ is the midpoint of $AB$, $D$ is the midpoint of $BC$, $E$ is the midpoint of $AD$, and $F$ is the midpoint of $BD$, find the length of segment $EF$. [b]p9.[/b] Vincent the Bug and Achyuta the Anteater are climbing an infinitely tall vertical bamboo stalk. Achyuta begins at the bottom of the stalk and climbs up at a rate of $5$ inches per second, while Vincent begins somewhere along the length of the stalk and climbs up at a rate of $3$ inches per second. After climbing for $t$ seconds, Achyuta is half as high above the ground as Vincent. Given that Achyuta catches up to Vincent after another $160$ seconds, compute $t$. [b]p10.[/b] What is the minimum possible value of $|x - 2022| + |x - 20|$ over all real numbers $x$? [b]p11.[/b] Let $ABCD$ be a rectangle. Lines $\ell_1$ and $\ell_2$ divide $ABCD$ into four regions such that $\ell_1$ is parallel to $AB$ and line $\ell_2$ is parallel to $AD$. Given that three of the regions have area $6$, $8$, and $12$, compute the sum of all possible areas of the fourth region. [b]p12.[/b] A diverse number is a positive integer that has two or more distinct prime factors. How many diverse numbers are less than $50$? [b]p13.[/b] Let $x$, $y$, and $z$ be real numbers so that $(x+y)(y +z) = 36$ and $(x+z)(x+y) = 4$. Compute $y^2 -x^2$. [b]p14.[/b] What is the remainder when $ 1^{10} + 3^{10} + 7^{10}$ is divided by $58$? [b]p15.[/b] Let $A = (0, 1)$, $B = (3, 5)$, $C = (1, 4)$, and $D = (3, 4)$ be four points in the plane. Find the minimum possible value of $AP + BP + CP + DP$ over all points $P$ in the plane. [b]p16.[/b] In trapezoid $ABCD$, points $E$ and $F$ lie on sides $BC$ and $AD$, respectively, such that $AB \parallel CD \parallel EF$. Given that $AB = 3$, $EF = 5$, and $CD = 6$, the ratio $\frac{[ABEF]}{[CDFE]}$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. (Note: $[F]$ denotes the area of $F$.) [b]p17.[/b] For sets $X$ and $Y$ , let $|X \cap Y |$ denote the number of elements in both $X$ and $Y$ and $|X \cup Y|$ denote the number of elements in at least one of $X$ or $Y$ . How many ordered pairs of subsets $(A,B)$ of $\{1, 2, 3,..., 8\}$ are there such that $|A \cap B| = 2$ and $|A \cup B| = 5$? [b]p18.[/b] A tetromino is a polygon composed of four unit squares connected orthogonally (that is, sharing a edge). A tri-tetromino is a polygon formed by three orthogonally connected tetrominoes. What is the maximum possible perimeter of a tri-tetromino? [b]p19.[/b] The numbers from $1$ through $2022$, inclusive, are written on a whiteboard. Every day, Hermione erases two numbers $a$ and $b$ and replaces them with $ab+a+b$. After some number of days, there is only one number $N$ remaining on the whiteboard. If $N$ has $k$ trailing nines in its decimal representation, what is the maximum possible value of $k$? [b]p20.[/b] Evaluate $5(2^2 + 3^2) + 7(3^2 + 4^2) + 9(4^2 + 5^2) + ... + 199(99^2 + 100^2)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A convex polygon with $2n$ sides is called [i]rhombic [/i] if its sides are equal and all pairs of opposite sides are parallel. A rhombic polygon can be partitioned into rhombic quadrilaterals. For what value of$ n$, a $2n$-sided rhombic polygon splits into $666$ rhombic quadrilaterals?

Let $n$ be a positive integer $\geq 2$ . Consider a $n$ by $n$ grid with all entries $1$. Define an operation on a square to be changing the signs of all squares adjacent to it but not the sign of its own. Find all $n$ such that it is possible after a finite sequence of operations to reach a $n$ by $n$ grid with all entries $-1$

Let $N\ge2$ be a fixed positive integer. There are $2N$ people, numbered $1,2,...,2N$, participating in a tennis tournament. For any two positive integers $i,j$ with $1\le i<j\le 2N$, player $i$ has a higher skill level than player $j$. Prior to the first round, the players are paired arbitrarily and each pair is assigned a unique court among $N$ courts, numbered $1,2,...,N$. During a round, each player plays against the other person assigned to his court (so that exactly one match takes place per court), and the player with higher skill wins the match (in other words, there are no upsets). Afterwards, for $i=2,3,...,N$, the winner of court $i$ moves to court $i-1$ and the loser of court $i$ stays on court $i$; however, the winner of court 1 stays on court 1 and the loser of court 1 moves to court $N$. Find all positive integers $M$ such that, regardless of the initial pairing, the players $2, 3, \ldots, N+1$ all change courts immediately after the $M$th round. [i]Proposed by Ray Li[/i]

Suppose there are $2019$ distinct points in a plane and the distances between pairs of them attain $k$ different values. Prove that $k$ is at least $44$.

Let $0<p<1$ be given. Initially, we have $n$ coins, all of which have probability $p$ of landing on heads, and probability $1-p$ of landing on tails (the results of the tosses are independent of each other). In each round, we toss our coins and remove those that result in heads. We keep repeating this until all our coins are removed. Let $k_n$ denote the expected number of rounds that are needed to get rid of all the coins. Prove that there exists $c>0$ for which the following inequality holds for all $n>0$ \[c\bigg(1+\frac{1}{2}+\cdots+\frac{1}{n}\bigg)<k_n<1+c\bigg(1+\frac{1}{2}+\cdots+\frac{1}{n}\bigg).\]

Consider a matrix of size $n\times n$ whose entries are real numbers of absolute value not exceeding $1$. The sum of all entries of the matrix is $0$. Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding $C$ in absolute value. [i]Proposed by Marcin Kuczma, Poland[/i]

The vertices of a regular $2n$-gon (with $n > 2$ an integer) are labelled with the numbers $1,2,...,2n$ in some order. Assume that the sum of the labels at any two adjacent vertices equals the sum of the labels at the two diametrically opposite vertices. Prove that this is possible if and only if $n$ is odd.

Let $N$ be the set of positive integers. If $A,B,C \ne \emptyset$, $A \cap B = B \cap C = C \cap A = \emptyset$ and $A \cup B \cup C = N$, we say that $A,B,C$ are partitions of $N$. Prove that there are no partitions of $N, A,B,C$, that satisfy the following: (i) $\forall a \in A, b \in B$, we have $a + b + 1 \in C$ (ii) $\forall b \in B, c \in C$, we have $b + c + 1 \in A$ (iii) $\forall c \in C, a \in A$, we have $c + a + 1 \in B$

The points of the plane are colored in black and white so that whenever three vertices of a parallelogram are the same color, the fourth vertex is that color, too. Prove that all the points of the plane are the same color.