On a circle, numbers from $1$ to $100$ are arranged in some order. We call a pair of numbers [i]good [/i] if these two numbers do not stand side by side, and at least on one of the two arcs into which they break a circle, all the numbers are less than each of them. What can be the total number of [i]good [/i] pairs?

We call a composite positive integer $n$ nice if it is possible to arrange its factors that are larger than $1$ on a circle such that two neighboring numbers are not coprime. How many of the elements of the set $\{1, 2, 3, ..., 100\}$ are nice?

Let $n$ and $k$ be positive integers. Given $n$ closed discs in the plane such that no matter how we choose $k + 1$ of them, there are always two of the chosen discs that have no common point. Prove that the $n$ discs can be partitioned into at most $10k$ classes such that any two discs in the same class have no common point.

On a $ 10 \times 10$ chessboard, some $ 4n$ unit squares are chosen to form a region $ \mathcal{R}.$ This region $ \mathcal{R}$ can be tiled by $ n$ $ 2 \times 2$ squares. This region $ \mathcal{R}$ can also be tiled by a combination of $ n$ pieces of the following types of shapes ([i]see below[/i], with rotations allowed). Determine the value of $ n.$

Let $S=\{1,2,3,\cdots,100\}$. Find the maximum value of integer $k$, such that there exist $k$ different nonempty subsets of $S$ satisfying the condition: for any two of the $k$ subsets, if their intersection is nonemply, then the minimal element of their intersection is not equal to the maximal element of either of the two subsets.

Suppose $n$ is a natural number. In how many ways can we place numbers $1,2,....,n$ around a circle such that each number is a divisor of the sum of it's two adjacent numbers?

Arranged in a circle are $2010$ digits, each of them equal to $1$, $2$, or $3$. For each positive integer $k$, it's known that in any block of $3k$ consecutive digits, each of the digits appears at most $k+10$ times. Prove that there is a block of several consecutive digits with the same number of $1$s, $2$s, and $3$s.

Consider an infinite square board with an integer written in each square. Assume that for each square the integer in it is equal to the sum of its neighbor to the left and its neighbor above. Assume also that there exists a row $R_0$ in the board such that all numbers in $R_0$ are positive. Denote by $R_1$ the row below $R_0$ , by $R_2$ the row below $R_1$ etc. Show that for each $N \ge 1$ the row $R_N$ cannot contain more than $N$ zeroes.

Ivan has a $n \times n$ board. He colors some of the squares black such that every black square has exactly two neighbouring square that are also black. Let $d_n$ be the maximum number of black squares possible, prove that there exist some real constants $a$, $b$, $c\ge 0$ such that; $$an^2-bn\le d_n\le an^2+cn.$$ [i]Proposed by Ivan Chan Kai Chin[/i]

Given a set $S$ that consists of $n \geq 3$ positive integers. It is known that if for some (not necessarily distinct) numbers $a,b,c,d$ from $S$ the equality $a-b=2(c-d)$ holds, then $a=b$ and $c=d$. Let $M$ be the biggest element in $S$. a) Prove that $M > \frac{n^2}{3}$. b) For $n=1024$ find the biggest possible value of $M$. [i]M. Zorka, Y. Sheshukou[/i]

Punches in the buses of a certain bus company always cut exactly six holes into the ticket. The possible locations of the holes form a $3 \times 3$ table as shown in the figure. Mr. Freerider wants to put together a collection of tickets such that, for any combination of punch holes, he would have a ticket with the same combination in his collection. The ticket can be viewed both from the front and from the back. Find the smallest number of tickets in such a collection. [img]https://cdn.artofproblemsolving.com/attachments/b/b/de5f09317a9a109fbecccecdc033de18217806.png[/img]

Define a domino to be a $1\times 2$ rectangular block. A $2023\times 2023$ square grid is filled with non-overlapping dominoes, leaving a single $1\times 1$ gap. John then repeatedly slides dominoes into the gap; each domino is moved at most once. What is the maximum number of times that John could have moved a domino? (Example: In the $3\times 3$ grid shown below, John could move 2 dominoes: $D$, followed by $A$.) [asy] unitsize(18); draw((0,0)--(3,0)--(3,3)--(0,3)--(0,0)--cycle); draw((0,1)--(3,1)); draw((2,0)--(2,3)); draw((1,1)--(1,3)); label("A",(0.5,2)); label("B",(1.5,2)); label("C",(2.5,2)); label("D",(1,0.5)); [/asy]

For what values of $ n$ does there exist an $ n \times n$ array of entries -1, 0 or 1 such that the $ 2 \cdot n$ sums obtained by summing the elements of the rows and the columns are all different?

At a strange party, each person knew exactly $22$ others. For any pair of people $X$ and $Y$ who knew each other, there was no other person at the party that they both knew. For any pair of people $X$ and $Y$ who did not know one another, there were exactly 6 other people that they both knew. How many people were at the party?

Alice and Bob are playing a game, starting with a binary string$ b$ of length $2022$. In each step, the rightmost digit of the string is deleted. If the deleted digit was $1$, Alice gets to choose which digit she wants to append on the left. Otherwise, Bob gets to choose the digit to append on the left of the string. Alice would like to turn the string $b$ into the all-zero string $\underbrace{00 . . . 0}_{2022}$, in the least number of steps possible, while Bob would like to maximize the number of steps necessary, or prevent Alice from doing this at all. a) Is there a string $b$ for which Bob can prevent Alice in her goal, if both players play optimally? b) If the answer to part a is yes, find all such strings $b$. If the answer is no, find the maximal game time and find the set of strings $b$ for which the game time is maximal.

In a club, there are $n$ members, and they are deciding some sport training sessions, satisfying all of these requirements: [i]i)[/i] Each member attends in all training sessions; [i]ii)[/i] At each training sessions, the club are divided into $3$ groups: swimming group, cycling group, running group (each member joins exactly $1$ group and each group consists of at least $1$ person); [i]iii)[/i] For any $2$ members of the club, we can find at least $1$ session such that there are $2$ people that are not in the same group. a) Assume that $n=9$. We know that the club has ran $1$ training session and it will run $1$ more session. How many ways to divide the group for the second training session? b) Assume that $n=2022$. Find the minimum number of training sessions that the club have to run?

Zscoder has an simple undirected graph $G$ with $n\ge 3$ vertices. Navi labels a positive integer to each vertex, and places a token at one of the vertex. This vertex is now marked red. In each turn, Zscoder plays with following rule: $\bullet$ If the token is currently at vertex $v$ with label $t$, then he can move the token along the edges in $G$ (possibly repeating some edges) exactly $t$ times. After these $t$ moves, he marks the current vertex red where the token is at if it is unmarked, or does nothing otherwise, then finishes the turn. Zscoder claims that he can mark all vertices in $G$ red after finite number of turns, regardless of Navi's labels and starting vertex. What is the minimum number of edges must $G$ have, in terms of $n$? [i]Proposed by Yeoh Zi Song[/i]

Let $\mathcal{C}$ be a family of subsets of $A=\{1,2,\dots,100\}$ satisfying the following two conditions: 1) Every $99$ element subset of $A$ is in $\mathcal{C}.$ 2) For any non empty subset $C\in\mathcal{C}$ there is $c\in C$ such that $C\setminus\{c\}\in \mathcal{C}.$ What is the least possible value of $|\mathcal{C}|$?

$n$ people met at the party, with $n \ge 2$. Each person dislikes exactly one other person present at the party (but not necessarily reciprocal, i.e. it may happen that $A$ dislikes $B$ even though $B$ does not dislike $A$) and likes all others. Prove that guests can be seated at three tables in such a way that each guest likes all the people at his table.

In an exotic country, the National Bank issues coins that can take any value in the interval $[0, 1]$. Find the smallest constant $c > 0$ such that the following holds, no matter the situation in that country: [i]Any citizen of the exotic country that has a finite number of coins, with a total value of no more than $1000$, can split those coins into $100$ boxes, such that the total value inside each box is at most $c$.[/i]

Let $a_1, a_2, ....., a_{2003}$ be sequence of reals number. Call $a_k$ $leading$ element, if at least one of expression $a_k; a_k+a_{k+1}; a_k+a_{k+1}+a_{k+2}; ....; a_k+a{k+1}+a_{k+2}+....+a_{2003}$ is positive. Prove, that if exist at least one $leading$ element, then sum of all $leading$'s elements is positive. Official solution [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here[/url]

[b]p1.[/b] $\vartriangle DEF$ is constructed from equilateral $\vartriangle ABC$ by choosing $D$ on $AB$, $E$ on $BC$ and $F$ on $CA$ so that $\frac{DB}{AB}=\frac{EC}{BC}=\frac{FA}{CA}=a$, where $a$ is a number between $0$ and $1/2$. (a) Show that $\vartriangle DEF$ is also equilateral. (b) Determine the value of $a$ that makes the area of $\vartriangle DEF$ equal to one half the area of $\vartriangle ABC$. [b]p2.[/b] A bowl contains some red balls and some white balls. The following operation is repeated until only one ball remains in the bowl: Two balls are drawn at random from the bowl. If they have different colors, then the red one is discarded and the white one is returned to the bowl. If they have the same color, then both are discarded and a red ball (from an outside supply of red balls) is added to the bowl. (Note that this operation—in either case—reduces the number of balls in the bowl by one.) (a) Show that if the bowl originally contained exactly $1$ red ball and $ 2$ white balls, then the color of the ball remaining at the end (i.e., after two applications of the operation) does not depend on chance, and determine the color of this remaining ball. (b) Suppose the bowl originally contained exactly $1986$ red balls and $1986$ white balls. Show again that the color of the ball remaining at the end does not depend on chance and determine its color. [b]p3.[/b] Let $a, b$, and $c$ be three consecutive positive integers, with $a < b < c.$ (a) Show that $ab$ cannot be the square of an integer. (b) Show that $ac$ cannot be the square of an integer. (c) Show that $abc$ cannot be the square of an integer. [b]p4.[/b] Consider the system of equations $$\sqrt{x}+\sqrt{y}=2$$ $$ x^2+y^2=5$$ (a) Show (algebraically or graphically) that there are two or more solutions in real numbers $x$ and $y$. (b) The graphs of the two given equations intersect in exactly two points. Find the equation of the straight line passing through these two points of intersection. [b]p5.[/b] Let $n$ and $m$ be positive integers. An $n \times m $ rectangle is tiled with unit squares. Let $r(n, m)$ denote the number of rectangles formed by the edges of these unit squares. Thus, for example, $r(2, 1) = 3$. (a) Find $r(2, 3)$. (b) Find $r(n, 1)$. (c) Find, with justification, a formula for $r(n, m)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A sub-graph of a complete graph with $n$ vertices is chosen such that the number of its edges is a multiple of $3$ and degree of each vertex is an even number. Prove that we can assign a weight to each triangle of the graph such that for each edge of the chosen sub-graph, the sum of the weight of the triangles that contain that edge equals one, and for each edge that is not in the sub-graph, this sum equals zero. [i]Proposed by Morteza Saghafian[/i]

A communications network consisting of some terminals is called a [i]$3$-connector[/i] if among any three terminals, some two of them can directly communicate with each other. A communications network contains a [i]windmill[/i] with $n$ blades if there exist $n$ pairs of terminals $\{x_{1},y_{1}\},\{x_{2},y_{2}\},\ldots,\{x_{n},y_{n}\}$ such that each $x_{i}$ can directly communicate with the corresponding $y_{i}$ and there is a [i]hub[/i] terminal that can directly communicate with each of the $2n$ terminals $x_{1}, y_{1},\ldots,x_{n}, y_{n}$ . Determine the minimum value of $f (n)$, in terms of $n$, such that a $3$ -connector with $f (n)$ terminals always contains a windmill with $n$ blades.

Let $G$ be a simple graph with 100 vertices such that for each vertice $u$, there exists a vertice $v \in N \left ( u \right )$ and $ N \left ( u \right ) \cap N \left ( v \right ) = \o $. Try to find the maximal possible number of edges in $G$. The $ N \left ( . \right )$ refers to the neighborhood.