The following operation is allowed on a finite graph: Choose an arbitrary cycle of length 4 (if there is any), choose an arbitrary edge in that cycle, and delete it from the graph. For a fixed integer ${n\ge 4}$, find the least number of edges of a graph that can be obtained by repeated applications of this operation from the complete graph on $n$ vertices (where each pair of vertices are joined by an edge). [i]Proposed by Norman Do, Australia[/i]

For any convex polygon $ W $ with area 1, let us denote by $ f(W) $ the area of the convex polygon whose vertices are the centers of all sides of the polygon $ W $. For each natural number $ n \geq 3 $, determine the lower limit and the upper limit of the set of numbers $ f(W) $ when $ W $ runs through the set of all $ n $ convex angles with area 1.

A card deck consists of $1024$ cards. On each card, a set of distinct decimal digits is written in such a way that no two of these sets coincide (thus, one of the cards is empty). Two players alternately take cards from the deck, one card per turn. After the deck is empty, each player checks if he can throw out one of his cards so that each of the ten digits occurs on an even number of his remaining cards. If one player can do this but the other one cannot, the one who can is the winner; otherwise a draw is declared. Determine all possible first moves of the first player after which he has a winning strategy. [i]Proposed by Ilya Bogdanov & Vladimir Bragin, Russia[/i]

A sequence $a_1, a_2, \ldots $ consisting of $1$'s and $0$'s satisfies for all $k>2016$ that \[ a_k=0 \quad \Longleftrightarrow \quad a_{k-1}+a_{k-2}+\cdots+a_{k-2016}>23. \] Prove that there exist positive integers $N$ and $T$ such that $a_k=a_{k+T}$ for all $k>N$.

Let $n$ be a natural number. Suppose $A$ and $B$ are two sets, each containing $n$ points in the plane, such that no three points of a set are collinear. Let $T(A)$ be the number of broken lines, each containing $n-1$ segments, and such that it doesn't intersect itself and its vertices are points of $A$. Define $T(B)$ similarly. If the points of $B$ are vertices of a convex $n$-gon (are in [i]convex position[/i]), but the points of $A$ are not, prove that $T(B)<T(A)$. [i]Proposed by Ali Khezeli[/i]

Let $n\geq1$ be an integer and let $t_1<t_2<\dots<t_n$ be positive integers. In a group of $t_n+1$ people, some games of chess are played. Two people can play each other at most once. Prove that it is possible for the following two conditions to hold at the same time: (i) The number of games played by each person is one of $t_1,t_2,\dots,t_n$. (ii) For every $i$ with $1\leq i\leq n$, there is someone who has played exactly $t_i$ games of chess.

On each side of a triangle, $5$ points are chosen (other than the vertices of the triangle) and these $15$ points are colored red. How many ways are there to choose four red points such that they form the vertices of a quadrilateral?

A pond has $2025$ lily pads arranged in a circle. Two frogs, Alice and Bob, begin on different lily pads. A frog jump is a jump which travels $2$, $3$, or $5$ positions clockwise. Alice and Bob each make a series of frog jumps, and each frog ends on the same lily pad that it started from. Given that each lily pad is the destination of exactly one jump, prove that each frog completes exactly two laps around the pond (i.e. travels $4050$ positions in total). [i]Linus Tang[/i]

$101$ blue and $101$ red points are selected on the plane, and no three lie on one straight line. The sum of the pairwise distances between the red points is $1$ (that is, the sum of the lengths of the segments with ends at red points), the sum of the pairwise distances between the blue ones is also $1$, and the sum of the lengths of the segments with the ends of different colors is $400$. Prove that you can draw a straight line separating everything red dots from all blue ones.

Suppose that every integer has been given one of the colours red, blue, green or yellow. Let $x$ and $y$ be odd integers so that $|x| \neq |y|$. Show that there are two integers of the same colour whose difference has one of the following values: $x,y,x+y$ or $x-y$.

The harmonic table is a triangular array: $1$ $\frac 12 \qquad \frac 12$ $\frac 13 \qquad \frac 16 \qquad \frac 13$ $\frac 14 \qquad \frac 1{12} \qquad \frac 1{12} \qquad \frac 14$ Where $a_{n,1} = \frac 1n$ and $a_{n,k+1} = a_{n-1,k} - a_{n,k}$ for $1 \leq k \leq n-1.$ Find the harmonic mean of the $1985^{th}$ row.

For $S_n=\{1, 2, ..., n\}$, find the maximum value of $m$ that makes the following proposition true. [b]Proposition[/b] There exists $m$ different subsets of $S$, say $A_1, A_2, ..., A_m$, such that for every $i, j=1, 2, ..., m$, the set $A_i \cup A_j$ is not $S$.

Let $n \ge 3$ be an odd integer. Each cell is a $n \times n$ board painted in yellow or blue. Let's call the sequence of cells $S_1, S_2,...,S_m$ [i]path [/i] if they are all the same color and the cells $S_i$ and $S_j$ have one in common an edge if and only if $|i - j| = 1$. Suppose that all yellow cells form a path and all the blue cells form a path. Prove that one of the two paths begins or ends at the center of the board.

A subset $B$ of $\{1, 2, \dots, 2017\}$ is said to have property $T$ if any three elements of $B$ are the sides of a nondegenerate triangle. Find the maximum number of elements that a set with property $T$ may contain.

Every positive integer is either [i]nice [/i] or [i]naughty[/i], and the Oracle of Numbers knows which are which. However, the Oracle will not directly tell you whether a number is [i]nice [/i] or [i]naughty[/i]. The only questions the Oracle will answer are questions of the form “What is the sum of all nice divisors of $n$?,” where $n$ is a number of the questioner’s choice. For instance, suppose ([i]just [/i] for this example) that $2$ and $3$ are nice, while $1$ and $6$ are [i]naughty[/i]. In that case, if you asked the Oracle, “What is the sum of all nice divisors of $6$?,” the Oracle’s answer would be $5$. Show that for any given positive integer $n$ less than $1$ million, you can determine whether $n$ is [i]nice [/i] or [i]naughty [/i] by asking the Oracle at most four questions.

Given $2018$ ones in a row: $$\underbrace{1\,\,\,1\,\,\,1\,\,\,1 \,\,\, ... \,\,\,1 \,\,\,1 \,\,\,1 \,\,\,1}_{2018 \,\,\, ones}$$ in which plus symbols $(+)$ are allowed to be inserted in between the ones. What is the maximum number of plus symbols $(+)$ that need to be inserted so that the resulting sum is 8102?

Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1, a_2, \ldots , a_n)$ and $(b_1, b_2, \ldots , b_n)$ is defined as $(a_1b_1, a_2b_2, \ldots , a_nb_n)$. Prove that there exists a sequence $(c_1, c_2, \ldots , c_n)$ such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1, c_2, \ldots , c_n)$ contains at most $\frac{k^2}{2^n}$ sequences.

In how many ways can we paint $16$ seats in a row, each red or green, in such a way that the number of consecutive seats painted in the same colour is always odd?

A game consists in $9$ coins (blacks or whites) arrenged in the following position (see picture 1). If you choose $1$ coin on the border of the square, this coin and it's neighbours change their color. If you choose the coin at the centre, it doesn't change it's color, but the other $8$ coins do. Here is an example of $9$ white coins, and the changes of their colors, choosing the coin said: (see picture 2). Is it possible, starting with $9$ white coins, to have $9$ black coins?.

Let $p$ and $q$ be two different primes. Prove that \[\left\lfloor\frac{p}{q}\right\rfloor+\left\lfloor\frac{2p}{q}\right\rfloor+\left\lfloor\frac{3p}{q}\right\rfloor+\ldots +\left\lfloor\frac{(q-1)p}{q}\right\rfloor=\frac{1}{2}(p-1)(q-1) \]

Consider a convex $15$- gon with perimeter $21$. Show that there one can select three distinct pairs of vertices that form a triangle with area less than $1$. [hide=original wording of second sentence]Zeige, dass man davon drei paarweise verschiedene Eckpunkte auswählen kann, die ein Dreieck mit Fläche kleiner als 1 bilden.[/hide]

There are $n$ people standing on a circular track. We want to perform a number of [i]moves[/i] so that we end up with a situation where the distance between every two neighbours is the same. The [i]move[/i] that is allowed consists in selecting two people and asking one of them to walk a distance $d$ on the circular track clockwise, and asking the other to walk the same distance on the track anticlockwise. The two people selected and the quantity $d$ can vary from move to move. Prove that it is possible to reach the desired situation (where the distance between every two neighbours is the same) after at most $n-1$ moves.

A board consists of $2021 \times 2021$ squares all of which are white, except for one corner square which is black. Alma and Bertha play the following game. At the beginning, there is a piece on the black square. In each turn, the player must move the piece to a new square in the same row or column as the one in which the piece is currently. All squares that the piece moves across, including the ending square but excluding the starting square, must be white, and all squares that the piece moves across, including the ending square, become black by this move. Alma begins, and the first player unable to move loses. Which player may prepare a strategy which secures her the victory? [img]https://cdn.artofproblemsolving.com/attachments/a/7/270d82f37b729bfe661f8a92cea8be67e5625c.png[/img]

Twenty-six people gather in a house. Alicia is friends with only one person, Bruno is friends with two people, Carlos is a friend of three, Daniel is four, Elías is five, and so following each person is friend of a person more than the previous person, until reaching Yvonne, the person number twenty-five, who is a friend to everyone. How many people is Zoila a friend of, person number twenty-six? Clarification: If $A$ is a friend of $B$ then $B$ is a friend of $A$.

Given a graph with $99$ vertices and degrees in $\{81,82,\dots,90\}$, prove that there exist $10$ vertices of this graph with equal degrees and a common neighbour. [i]Proposed by Alireza Alipour[/i]