Let $P$ and $P '$ be two unequal regular $n-$gons and $A$ and $A'$two points inside $P$ and$ P '$, respectively.Suppose $\{ d_1 , d_2 , \cdots d_n \}$ are the distances from $A $ to the vertices of $P$ and $\{ d'_1 , d'_2 , \cdots d'_n \}$ are defines similarly for $P',A'$. Is it possible for $\{ d'_1 , d'_2 , \cdots d'_n \}$ to be a permutation of $\{ d_1 , d_2 , \cdots d_n \}$ ?

In the city of Flensburg there is a single, infinitely long, street with housesnumbered $2, 3, \ldots$. The police in Flensburg is trying to catch a thief who every night moves from the house where she is currently hiding to one of its neighbouring houses. To taunt the local law enforcement the thief reveals every morning the highest prime divisor of the number of the house she has moved to. Every Sunday afternoon the police searches a single house, and they catch the thief if they search the house she is currently occupying. Does the police have a strategy to catch the thief in finite time?

Five pairs of cartoon characters, Donald and Katrien Duck, Asterix and Obelix, Suske and Wiske, Tom and Jerry, Heer Bommel and Tom Poes, sit around a round table with $10$ chairs. The two members of each pair ensure that they sit next to each other. In how many different ways can the ten seats be occupied? Two ways are different if they cannot be transferred to each other by a rotation.

[b]p1.[/b] Two robots race on the plane from $(0, 0)$ to $(a, b)$, where $a$ and $b$ are positive real numbers with $a < b$. The robots move at the same constant speed. However, the first robot can only travel in directions parallel to the lines $x = 0$ or $y = 0$, while the second robot can only travel in directions parallel to the lines $y = x$ or $y = -x$. Both robots take the shortest possible path to $(a, b)$ and arrive at the same time. Find the ratio $\frac{a}{b}$ . [b]p2.[/b] Suppose $x + \frac{1}{x} + y + \frac{1}{y} = 12$ and $x^2 + \frac{1}{x^2} + y^2 + \frac{1}{y^2} = 70$. Compute $x^3 + \frac{1}{x^3} + y^3 + \frac{1}{y^3}$. [b]p3.[/b] Find the largest non-negative integer $a$ such that $2^a$ divides $$3^{2^{2018}}+ 3.$$ [b]p4.[/b] Suppose $z$ and $w$ are complex numbers, and $|z| = |w| = z \overline{w}+\overline{z}w = 1$. Find the largest possible value of $Re(z + w)$, the real part of $z + w$. [b]p5.[/b] Two people, $A$ and $B$, are playing a game with three piles of matches. In this game, a move consists of a player taking a positive number of matches from one of the three piles such that the number remaining in the pile is equal to the nonnegative difference of the numbers of matches in the other two piles. $A$ and $B$ each take turns making moves, with $A$ making the first move. The last player able to make a move wins. Suppose that the three piles have $10$, $x$, and $30$ matches. Find the largest value of $x$ for which $A$ does not have a winning strategy. [b]p6.[/b] Let $A_1A_2A_3A_4A_5A_6$ be a regular hexagon with side length $1$. For $n = 1$,$...$, $6$, let $B_n$ be a point on the segment $A_nA_{n+1}$ chosen at random (where indices are taken mod $6$, so $A_7 = A_1$). Find the expected area of the hexagon $B_1B_2B_3B_4B_5B_6$. [b]p7.[/b] A termite sits at the point $(0, 0, 0)$, at the center of the octahedron $|x| + |y| + |z| \le 5$. The termite can only move a unit distance in either direction parallel to one of the $x$, $y$, or $z$ axes: each step it takes moves it to an adjacent lattice point. How many distinct paths, consisting of $5$ steps, can the termite use to reach the surface of the octahedron? [b]p8.[/b] Let $$P(x) = x^{4037} - 3 - 8 \cdot \sum^{2018}_{n=1}3^{n-1}x^n$$ Find the number of roots $z$ of $P(x)$ with $|z| > 1$, counting multiplicity. [b]p9.[/b] How many times does $01101$ appear as a not necessarily contiguous substring of $0101010101010101$? (Stated another way, how many ways can we choose digits from the second string, such that when read in order, these digits read $01101$?) [b]p10.[/b] A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. For example, $28$ is a perfect number because $1 + 2 + 4 + 7 + 14 = 28$. Let $n_i$ denote the ith smallest perfect number. Define $$f(x) =\sum_{i|n_x}\sum_{j|n_i}\frac{1}{j}$$ (where $\sum_{i|n_x}$ means we sum over all positive integers $i$ that are divisors of $n_x$). Compute $f(2)$, given there are at least $50 $perfect numbers. [b]p11.[/b] Let $O$ be a circle with chord $AB$. The perpendicular bisector to $AB$ is drawn, intersecting $O$ at points $C$ and $D$, and intersecting $AB$ at the midpoint $E$. Finally, a circle $O'$ with diameter $ED$ is drawn, and intersects the chord $AD$ at the point $F$. Given $EC = 12$, and $EF = 7$, compute the radius of $O$. [b]p12.[/b] Suppose $r$, $s$, $t$ are the roots of the polynomial $x^3 - 2x + 3$. Find $$\frac{1}{r^3 - 2}+\frac{1}{s^3 - 2}+\frac{1}{t^3 - 2}.$$ [b]p13.[/b] Let $a_1$, $a_2$,..., $a_{14}$ be points chosen independently at random from the interval $[0, 1]$. For $k = 1$, $2$,$...$, $7$, let $I_k$ be the closed interval lying between $a_{2k-1}$ and $a_{2k}$ (from the smaller to the larger). What is the probability that the intersection of $I_1$, $I_2$,$...$, $I_7$ is nonempty? [b]p14.[/b] Consider all triangles $\vartriangle ABC$ with area $144\sqrt3$ such that $$\frac{\sin A \sin B \sin C}{ \sin A + \sin B + \sin C}=\frac14.$$ Over all such triangles $ABC$, what is the smallest possible perimeter? [b]p15.[/b] Let $N$ be the number of sequences $(x_1,x_2,..., x_{2018})$ of elements of $\{1, 2,..., 2019\}$, not necessarily distinct, such that $x_1 + x_2 + ...+ x_{2018}$ is divisible by $2018$. Find the last three digits of $N$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The numbers $1, 2, 3, \ldots , 10$ are written on the blackboard. In each step, Andrew chooses two numbers $a, b$ which are written on the blackboard such that $a\geqslant 2b$, he erases them, and in their place writes the number $a-2b$. Find all numbers $n$, such that after a sequence of steps as above, at the end only the number $n$ will remain on the blackboard.

Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties: [list=disc] [*]every term in the sequence is less than or equal to $2^{2023}$, and [*]there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\] [/list]

Positive integers 1 to 9 are written in each square of a $ 3 \times 3 $ table. Let us define an operation as follows: Take an arbitrary row or column and replace these numbers $ a, b, c$ with either non-negative numbers $ a-x, b-x, c+x $ or $ a+x, b-x, c-x$, where $ x $ is a positive number and can vary in each operation. (1) Does there exist a series of operations such that all 9 numbers turn out to be equal from the following initial arrangement a)? b)? \[ a) \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} )\] \[ b) \begin{array}{ccc} 2 & 8 & 5 \\ 9 & 3 & 4 \\ 6 & 7 & 1 \end{array} )\] (2) Determine the maximum value which all 9 numbers turn out to be equal to after some steps.

Find the largest number $n$ that for which there exists $n$ positive integers such that non of them divides another one, but between every three of them, one divides the sum of the other two. [i]Proposed by Morteza Saghafian[/i]

King Arthur one day had to fight the Dragon with Three Heads and Three Tails. His task became easier when he managed to find a magic sword that could, with a single blow, do one (and only one) of the following things: $\bullet$ cut off a head, $\bullet$ cut off two heads, $\bullet$ cut a tail, $\bullet$ cut off two tails. Furthermore, Fairy Morgana revealed to him the dragon's secret: $\bullet$ if a head is cut off, a new one grows, $\bullet$ if two heads are cut off nothing happens, $\bullet$ in place of a tail, two new tails are born, $\bullet$ if two tails are cut off a new head grow, $\bullet$ and the dragon dies if it loses its three heads and three tails. How many hits are needed to kill the dragon?

Let $n$ be a positive integer. There are $n$ islands with $n-1$ bridges connecting them such that one can travel from any island to another. One afternoon, a fire breaks out in one of the islands. Every morning, it spreads to all neighbouring islands. (Two islands are neighbours if they are connected by a bridge.) To control the spread, one bridge is destroyed every night until the fire has nowhere to spread the next day. Let $X$ be the minimum possible number of bridges one has to destroy before the fire stops spreading. Find the maximum possible value of $X$ over all possible configurations of bridges and island where the fire starts at.

Suppose that for every $n$ the number $m(n)$ is chosen such that $m(n)\ln(m(n))=n-\frac 12$. Show that $b_n$ is asymptotic to the following expression where $b_n$ is the $n-$th Bell number, that is the number of ways to partition $\{1,2,\ldots,n\}$: \[ \frac{m(n)^ne^{m(n)-n-\frac 12}}{\sqrt{\ln n}}. \] Two functions $f(n)$ and $g(n)$ are asymptotic to each other if $\lim_{n\rightarrow \infty}\frac{f(n)}{g(n)}=1$.

Take $r$ such that $1\le r\le n$, and consider all subsets of $r$ elements of the set $\{1,2,\ldots,n\}$. Each subset has a smallest element. Let $F(n,r)$ be the arithmetic mean of these smallest elements. Prove that: \[ F(n,r)={n+1\over r+1}. \]

Rectangle $p*q,$ where $p,q$ are relatively coprime positive integers with $p <q$ is divided into squares $1*1$.Diagonal which goes from lowest left vertice to highest right cuts triangles from some squares.Find sum of perimeters of all such triangles.

We attempt to cover the plane with an infinite sequence of rectangles, overlapping allowed. (a) Is the task always possible if the area of the $n$th rectangle is $n^2$ for each $n$? (b) Is the task always possible if each rectangle is a square, and for any number $N$, there exist squares with total area greater than $N$?

Rodolfo and Gabriela have $9$ chips numbered from $1$ to $9$ and they have fun with the following game: They remove the chips one by one and alternately (until they have $3$ chips each), with the following rules: $\bullet$ Rodolfo begins the game, choosing a chip and in the following moves he must remove, each time, a chip three units greater than the last chip drawn by Gabriela. $\bullet$ Gabriela, on her turn, chooses a first chip and in the following times she must draw, each time, a chip two units smaller than the last chip that she herself drew. $\bullet$ The game is won by whoever gets the highest number by adding up their three tokens. $\bullet$ If the game cannot be completed, a tie is declared. If they play without making mistakes, how should Rodolfo play to be sure he doesn't lose?

Consider a tree with $n$ vertices, labeled with $1,\ldots,n$ in a way that no label is used twice. We change the labeling in the following way - each time we pick an edge that hasn't been picked before and swap the labels of its endpoints. After performing this action $n-1$ times, we get another tree with its labeling a permutation of the first graph's labeling. Prove that this permutation contains exactly one cycle.

There are $68$ coins , each coin having a different weight than that of each other . Show how to find the heaviest and lightest coin in $100$ weighings on a balance beam. (S. Fomin, Leningrad)

The game [i]Clobber [/i] is played by two on a strip of $2k$ squares. At the beginning there is a piece on each square, the pieces of both players stand alternatingly. At each move the player shifts one of his pieces to the neighbouring square that holds a piece of his opponent and removes his opponent’s piece from the table. The moves are made in turn, the player whose opponent cannot move anymore is the winner. Prove that if for some $k$ the player who does not start the game has the winning strategy, then for $k + 1$ and $k + 2$ the player who makes the first move has the winning strategy.

Joe finally asked Kathryn out. They go out on a date on a Friday night, racing at the local go-kart track. They take turns racing across an $8 \times 8$ square grid composed of $64$ unit squares. If Joe and Kathryn start in the lower left-hand corner of the $8\times 8$ square, and can move either up or right along any side of any unit square, what is the probability that Joe and Kathryn take the same exact path to reach the upper right-hand corner of the $8\times 8$ square grid?

Five elders are sitting around a large bonfire. They know that Oluf will put a hat of one of four colours (red, green, blue or yellow) on each elder’s head, and after a short time for silent reflection each elder will have to write down one of the four colours on a piece of paper. Each elder will only be able to see the colour of their two neighbours’ hats, not that of their own nor that of the remaining two elders’ hats, and they also cannot communicate after Oluf starts putting the hats on. Show that the elders can devise a strategy ahead of time so that at most two elders will end up writing down the colour of their own hat

Let $N$ be the number of non-empty subsets $T$ of $S = \{1,2, 3,4,...,2020\}$ satisfying $max (T) >1000$. Compute the largest integer $k$ such that $3^k$ divides $N$.

Five students $ A, B, C, D, E$ took part in a contest. One prediction was that the contestants would finish in the order $ ABCDE$. This prediction was very poor. In fact, no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction had the contestants finishing in the order $ DAECB$. This prediction was better. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.

There are $2020$ positive integers written on a blackboard. Every minute, Zuming erases two of the numbers and replaces them by their sum, difference, product, or quotient. For example, if Zuming erases the numbers $6$ and $3$, he may replace them with one of the numbers in the set $\{6+3, 6-3, 3-6, 6\times 3, 6\div 3, 3\div 6\}$ $= \{9, 3, 3, 18, 2, \tfrac 12\}$. After $2019$ minutes, Zuming writes the single number $-2020$ on the blackboard. Show that it was possible for Zuming to have ended up with the single number $2020$ instead, using the same rules and starting with the same $2020$ integers. [i]Proposed by Zhuo Qun (Alex) Song[/i]

A horse stands at the corner of a chessboard, a white square. With each jump, the horse can move either two squares horizontally and one vertically or two vertically and one horizontally (like a knight moves). The horse earns two carrots every time it lands on a black square, but it must pay a carrot in rent to rabbit who owns the chessboard for every move it makes. When the horse reaches the square on which it began, it can leave. What is the maximum number of carrots the horse can earn without touching any square more than twice? [img]https://cdn.artofproblemsolving.com/attachments/e/c/c817d92ead6cfb3868f9cb526fb4e1fd7ffe4d.png[/img]

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]