Let be three finite and nonempty sets $ A,B,C $ such that $ |A|=|C|\le |B| , $ and a bijection $ A\stackrel{\beta }{\longrightarrow } C. $ How many pairs of functions $ A\stackrel{f_2 }{\longrightarrow } B\stackrel{f_1 }{\longrightarrow } C $ that satisfy $ f_1\circ f_2=\beta $ are there?

[u]Round 5[/u] [b]p13.[/b] Two concentric circles have radii $1$ and $3$. Compute the length of the longest straight line segment that can be drawn froma point on the circle of radius $1$ to a point on the circle of radius $3$ if the segment cannot intersect the circle of radius $1$. [b]p14.[/b] Find the value of $\frac{1}{3} + \frac29+\frac{3}{27}+\frac{4}{81}+\frac{5}{243}+...$ [b]p15.[/b] Bob is trying to type the word "welp". However, he has a $18$ chance ofmistyping each letter and instead typing one of four adjacent keys, each with equal probability. Find the probability that he types "qelp" instead of "welp". [u]Round 6[/u] [b]p16.[/b] How many ways are there to tile a $2\times 12$ board using an unlimited supply of $1\times 1$ and $1\times 3$ pieces? [b]p17.[/b] Jeffrey and Yiming independently choose a number between $0$ and $1$ uniformly at random. What is the probability that their two numbers can formthe sidelengths of a triangle with longest side of length $1$? [b]p18.[/b] On $\vartriangle ABC$ with $AB = 12$ and $AC = 16$, let $M$ be the midpoint of $BC$ and $E$,$F$ be the points such that $E$ is on $AB$, $F$ is on $AC$, and $AE = 2AF$. Let $G$ be the intersection of $EF$ and $AM$. Compute $\frac{EG}{GF}$ . [u]Round 7[/u] [b]p19.[/b] Find the remainder when $2019x^{2019} -2018x^{2018}+ 2017x^{2017}-...+x$ is divided by $x +1$. [b]p20.[/b] Parallelogram $ABCD$ has $AB = 5$, $BC = 3$, and $\angle ABC = 45^o$. A line through C intersects $AB$ at $M$ and $AD$ at $N$ such that $\vartriangle BCM$ is isosceles. Determine the maximum possible area of $\vartriangle MAN$. [b]p21[/b]. Determine the number of convex hexagons whose sides only lie along the grid shown below. [img]https://cdn.artofproblemsolving.com/attachments/2/9/93cf897a321dfda282a14e8f1c78d32fafb58d.png[/img] [u]Round 8[/u] [b]p22.[/b] Let $\vartriangle ABC$ be a triangle with side lengths $AB = 4$, $BC = 5$, and $C A = 6$. Extend ray $\overrightarrow{AB}$ to a point $D$ such that $AD = 12$, and similarly extend ray $\overrightarrow{CB}$ to point $E$ such that $CE = 15$. Let $M$ and $N$ be points on the circumcircles of $ABC$ and $DBE$, respectively, such that line $MN$ is tangent to both circles. Determine the length of $MN$. [b]p23.[/b] A volcano will erupt with probability $\frac{1}{20-n}$ if it has not erupted in the last $n$ years. Determine the expected number of years between consecutive eruptions. [b]p24.[/b] If $x$ and $y$ are integers such that $x+ y = 9$ and $3x^2+4x y = 128$, find $x$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3165997p28809441]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166099p28810427]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$2n-1$ distinct positive real numbers with sum $S $ are given. Prove that there are at least $\binom {2n-2}{n-1}$ different ways to choose $n $ numbers among them such that their sum is at least $\frac {S}{2}$. [i]Proposed by Amirhossein Gorzi[/i]

There are $n$ blue points and $n$ red points on a straight line. Prove that the sum of all distances between pairs of points of the same colour is less than or equal to the sum of all distances between pairs of points of different colours

Let $n$ be a positive integer. Find the number of sequences $a_0,a_1,a_2,\dots,a_{2n}$ of integers in the range $[0,n]$ such that for all integers $0\leq k\leq n$ and all nonnegative integers $m$, there exists an integer $k\leq i\leq 2k$ such that $\lfloor k/2^m\rfloor=a_i.$ [i]Andrew Carratu[/i]

Given any positive real number $\varepsilon$, prove that, for all but finitely many positive integers $v$, any graph on $v$ vertices with at least $(1+\varepsilon)v$ edges has two distinct simple cycles of equal lengths. (Recall that the notion of a simple cycle does not allow repetition of vertices in a cycle.) [i]Fedor Petrov, Russia[/i]

For what values o f $n$ is it possible to place the integers from $1$ to $n$ inclusive on a circle (not necessarily in order) so that the sum of any two successive integers in the circle is divisible by the next one in the clockwise order? (A Shapovalov)

A positive integer $n$ is [i]Danish[/i] if a regular hexagon can be partitioned into $n$ congruent polygons. Prove that there are infinitely many positive integers $n$ such that both $n$ and $2^n+n$ are Danish.

In a group of interpreters each one speaks one of several foreign languages, 24 of them speak Japanese, 24 Malaysian, 24 Farsi. Prove that it is possible to select a subgroup in which exactly 12 interpreters speak Japanese, exactly 12 speak Malaysian and exactly 12 speak Farsi.

For any integer $n\geq 3$, let $A\left(n\right)$ denote the maximal number of self-intersections a closed broken line $P_1P_2...P_nP_1$ can have; hereby, we assume that no three vertices of the broken line $P_1P_2...P_nP_1$ are collinear. Prove that [b](a)[/b] if n is odd, then $A\left(n\right)=\frac{n\left(n-3\right)}{2}$; [b](b)[/b] if n is even, then $A\left(n\right)=\frac{n\left(n-4\right)}{2}+1$. [i]Note.[/i] A [i]self-intersection[/i] of a broken line is a (non-ordered) pair of two distinct non-adjacent segments of the broken line which have a common point.

Let \( n \geq 2 \) be an integer. Two players, Alice and Bob, play the following game on the complete graph \( K_n \): They take turns to perform operations, where each operation consists of coloring one or two edges that have not been colored yet. The game terminates if at any point there exists a triangle whose three edges are all colored. Prove that there exists a positive number \(\varepsilon\), Alice has a strategy such that, no matter how Bob colors the edges, the game terminates with the number of colored edges not exceeding \[ \left( \frac{1}{4} - \varepsilon \right) n^2 + n. \]

Does there exist a bijection $f:\mathbb{N}^{+} \rightarrow \mathbb{N}^{+}$, such that there exist a positive integer $k$, and it's possible to have each positive integer colored by one of $k$ chosen colors, such that for any $x \neq y$ , $f(x)+y$ and $f(y)+x$ are not the same color?

Let $n$ be a positive integer. In an $n\times4$ table, each row is equal to \[\begin{tabular}{| c | c | c | c |} \hline 2 & 0 & 1 & 0 \\ \hline \end{tabular}\] A [i]change[/i] is taking three consecutive boxes in the same row with different digits in them and changing the digits in these boxes as follows: \[0\to1\text{, }1\to2\text{, }2\to0\text{.}\] For example, a row $\begin{tabular}{| c | c | c | c |}\hline 2 & 0 & 1 & 0 \\ \hline\end{tabular}$ can be changed to the row $\begin{tabular}{| c | c | c | c |}\hline 0 & 1 & 2 & 0 \\ \hline\end{tabular}$ but not to $\begin{tabular}{| c | c | c | c |}\hline 2 & 1 & 2 & 1 \\ \hline\end{tabular}$ because $0$, $1$, and $0$ are not distinct. Changes can be applied as often as wanted, even to items already changed. Show that for $n<12$, it is not possible to perform a finite number of changes so that the sum of the elements in each column is equal.

Alice and Bob play a game on a numbered row of $n \ge 5$ squares. At the beginning a pebble is put on the first square and then the players make consecutive moves; Alice starts. During a move a player is allowed to choose one of the following: [list] [*] move the pebble one square forward; [*] move the pebble four squares forward; [*] move the pebble two squares backwards. [/list] All of the possible moves are only allowed if the pebble stays within the borders of the square row. The player who moves the pebble to the last square (a.k.a $n\text{-th}$) wins. Determine for which values of $n$ each of the players has a winning strategy.

Given a permutation $\sigma = (a_1,a_2,a_3,...a_n)$ of $(1,2,3,...n)$ , an ordered pair $(a_j,a_k)$ is called an inversion of $\sigma$ if $a \leq j < k \leq n$ and $a_j > a_k$. Let $m(\sigma)$ denote the no. of inversions of the permutation $\sigma$. Find the average of $m(\sigma)$ as $\sigma$ varies over all permutations.

A set of points in the plane is called [i]free [/i] if no three points of the set are the vertices of an equilateral triangle. Prove that any set of $n$ points in the plane has a free subset of at least $\sqrt{n}$ points

Real numbers are written on the squares of an infinite grid. Two figures consisting of finitely many squares are given. They may be translated anywhere on the grid as long as their squares coincide with those of the grid. It is known that wherever the first figure is translated, the sum of numbers it covers is positive. Prove that the second figure can be translated so that the sum of the numbers it covers is also positive.

For an integer $n>2$, the tuple $(1, 2, \ldots, n)$ is written on a blackboard. On each turn, one can choose two numbers from the tuple such that their sum is a perfect square and swap them to obtain a new tuple. Find all integers $n > 2$ for which all permutations of $\{1, 2,\ldots, n\}$ can appear on the blackboard in this way.

Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively. Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes. [i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]

I have an $n \times n$ sheet of stamps, from which I've been asked to tear out blocks of three adjacent stamps in a single row or column. (I can only tear along the perforations separating adjacent stamps, and each block must come out of the sheet in one piece.) Let $b(n)$ be the smallest number of blocks I can tear out and make it impossible to tear out any more blocks. Prove that there are real constants $c$ and $d$ such that \[ \dfrac{1}{7} n^2 - cn \leq b(n) \leq \dfrac{1}{5} n^2 + dn \] for all $n > 0$.

Let $p_n$ denote the probability that, in $n$ tosses, a fair coin shows the head up $100$ consecutive times. Prove that the sequence $(p_n)$ converges and determine its limit.

Let $N$ be a given positive integer. Consider a permutation of $1,2,3,\cdots,N$, denoted as $p_1,p_2,\cdots,p_N$. For a section $p_l, p_{l+1},\cdots, p_r$, we call it "extreme" if $p_l$ and $p_r$ are the maximum and minimum value of that section. We say a permutation $p_1,p_2,\cdots,p_N$ is "super balanced" if there isn't an "extreme" section with a length at least $3$. For example, $1,4,2,3$ is "super balanced", but $3,1,2,4$ isn't. Please answer the following questions: 1. How many "super balanced" permutations are there? 2. For each integer $M\leq N$. How many "super balanced" permutations are there such that $p_1=M$? [i]Proposed by ltf0501[/i]

In an $8 \times 8$ grid, $n$ disks, numbered $1$ to $n$ are stacked, with random order, in a pile in the bottom left comer. The disks can be moved one at a time to a neighbouring cell either to the right or top. The aim to move all the disks to the cell at the top right comer and stack them in the order $1,2,...,n$ from the bottom. Each cell, except the bottom left and top right cell, can have at most one disk at any given time. Find the largest value of $n$ so that the aim can be achieved.

Given a number $n\in\mathbb{Z}^+$ and let $S$ denotes the set $\{0,1,2,...,2n+1\}$. Consider the function $f:\mathbb{Z}\times S\to [0,1]$ satisfying two following conditions simultaneously: i) $f(x,0)=f(x,2n+1)=0\forall x\in\mathbb{Z}$; ii) $f(x-1,y)+f(x+1,y)+f(x,y-1)+f(x,y+1)=1$ for all $x\in\mathbb{Z}$ and $y\in\{1,2,3,...,2n\}$. Let $F$ be the set of such functions. For each $f\in F$, let $v(f)$ be the set of values of $f$. a) Proof that $|F|=\infty$. b) Proof that for each $f\in F$ then $|v(f)|<\infty$. c) Find the maximum value of $|v(f)|$ for $f\in F$.

Every unit square of a $n \times n$ board is colored either red or blue so that among all 2 $\times 2$ squares on this board all possible colorings of $2 \times 2$ squares with these two colors are represented (colorings obtained from each other by rotation and reflection are considered different). a) Find the least possible value of $n$. b) For the least possible value of $n$ find the least possible number of red unit squares