Two persons are playing the following game on a $n\times m$ table, with drawn lines: Person $\#1$ starts the game. Each person in their move, folds the table on one of its lines. The one that could not fold the table on their turn loses the game. Who has a winning strategy?

Two 100-digit binary sequences are given. In one operation, one may insert (possibly at the beggining or end) or remove one or more identical digits from a sequence. What is the smallest $k{}$ for which we can transform the first sequence into the second one in no more than $k{}$ operations? [i]Proposed by V. Novikov[/i]

The director of a Zoo has bought eight elephants numbered by $1, 2, \ldots , 8$. He has forgotten their masses but he remembers that each elephant starting with the third one has the mass equal to the sum of the masses of two preceding ones. Suddenly the director hears a rumor that one of the elephants has lost his mass. How can the director perform two weightings on balancing scales without weights to either find this elephant or make sure that this was just a rumor? (It is known that no elephant gained mass and no more than one elephant lost mass.) [i]Alexandr Gribalko[/i]

In a $3$ by $3$ table, by a $k$-worm, we mean a path of different cells $(S_1,S_2,...,S_k)$ such that each two consecutive cells have one side in common. The $k$-worm at each steep can go one cell forward and turn to the $(S,S_1,...,S_{k-1})$ if $S$ is an unfilled cell which is adjacent (has one side in common) with $S_1$. Find the maximum number of $k$ such that there is a $k$-worm $(S_1,...,S_k)$ such that after finitly many steps can be turned to $(S_k,...,S_1)$.

One hundred musicians are planning to organize a festival with several concerts. In each concert, while some of the one hundred musicians play on stage, the others remain in the audience assisting to the players. What is the least number of concerts so that each of the musicians has the chance to listen to each and every one of the other musicians on stage?

A dragon gave a captured knight $100$ coins. Half of them are magical, but only dragon knows which are. Each day, the knight should divide the coins into two piles (not necessarily equal in size). The day when either magic coins or usual coins are spread equally between the piles, the dragon set the knight free. Can the knight guarantee himself a freedom in at most (a) $50$ days? (b) $25$ days?

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]

The natural numbers $k \geqslant n$ are given. Peter has $n{}$ objects and $N{}$ special ways in which he likes to lay them out in a row from left to right. He noticed that for any non-empty subset $A{}$ of these objects containing $|A| \leqslant k$ objects, and any element $a\in A$, there are exactly $N/|A|$ special ways for which element $a{}$ is the leftmost in the set $A{}$. Prove that, under the same conditions on $A{}$ and $a{}$, for any integer $m =1,2,\ldots,|A|$ there are exactly $N/|A|$ special ways for which the element $a{}$ is the $m^{\text{th}}$ from the left in the set $A{}$.

Lucky and Jinx were given a paper with $2023$ points arranged as the vertices of a regular polygon. They were then tasked to color all the segments connecting these points such that no triangle formed with these points has all edges in the same color, nor in three different colors and no quadrilateral (not necessarily convex) has all edges in the same color. After the coloring it was determined that Jinx used at least two more colors than Lucky. How many colors did each of them use? [i]Authored by Ilija Jovcheski[/i]

Let $n\geq 3$ be an integer. Each vertex of a regular $n$-gon is labelled with a real number not exceeding $1$. For real numbers $a,b,c$ on any three consecutive vertices which are arranged clockwise in such an order, we have $c=|a-b|$. Determine the maximum value of the sum of all numbers in terms of $n$.

Let $ n \geq 3$ and consider a set $ E$ of $ 2n \minus{} 1$ distinct points on a circle. Suppose that exactly $ k$ of these points are to be colored black. Such a coloring is [b]good[/b] if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly $ n$ points from $ E$. Find the smallest value of $ k$ so that every such coloring of $ k$ points of $ E$ is good.

An $n\times n$ square grid ($n\geqslant 3$) is rolled into a cylinder. Some of the cells are then colored black. Show that there exist two parallel lines (horizontal, vertical or diagonal) of cells containing the same number of black cells. [i]E. Poroshenko[/i]

In a school with 5 different grades there are 250 girls and 250 boys. Each grade has the same number of students. for a competition of knowledge wants to form teams of a female and a male who are of the same grade. If in each grade there are at least $19$ females and $19$ males. Find the greatest amount of teams that can be formed.

Let $\,S\,$ be a finite set of points in three-dimensional space. Let $\,S_{x},\,S_{y},\,S_{z}\,$ be the sets consisting of the orthogonal projections of the points of $\,S\,$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that \[ \vert S\vert^{2}\leq \vert S_{x} \vert \cdot \vert S_{y} \vert \cdot \vert S_{z} \vert, \] where $\vert A \vert$ denotes the number of elements in the finite set $A$. [hide="Note"] Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane. [/hide]

You have a list of $2023$ numbers, where each one can be $-1$, $0$, $1$ or $2$. The sum of all numbers is $19$ and the sum of their squares is $99$. What are the minimum and maximum values of the sum of the cubes of those $2023$ numbers?

For each subset $T$ of $S = \{1, 2, ... , 7\}$, the result $r(T)$ of T is computed as follows: the elements of $T$ are written, largest to smallest, and alternating signs $(+, -)$ starting with $+$ are put in front of each number. The value of the resulting expression is$ r(T)$. (For example, for $T =\{2, 4, 7\}$, we have $r(T) = +7 - 4 + 2 = 5$.) Compute the sum of $r(T)$ as $T$ ranges over all subsets of $S$.

$n$ players are playing a game. Each player has $n$ tokens. Every turn, two players with at least one token are randomly selected. The player with less tokens gives one token to the player with more tokens. If both players have the same number of tokens, a coin flip decides which player receives a token and which player gives a token. The game ends when one player has all the tokens. If $n = 2019$, suppose the maximum number of turns the game could take to end can be written as $\frac{1}{d} (a \cdot 2019^3 + b \cdot 2019^2 + c \cdot 2019)$ for integers $a, b, c, d$. Find $\frac{abc}{d}$ .

Three cards, only one of which is an ace, are placed face down on a table. You select one, but do not look at it. The dealer turns over one of the other cards, which is not the ace (if neither are, he picks one of them randomly to turn over). You get a chance to change your choice and pick either of the remaining two face-down cards. If you selected the cards so as to maximize the chance of finding the ace on the second try, what is the probability that you selected it on the (a) first try? (b) second try?

Let $f$ be a function on non-negative integers defined as follows $$f(2n)=f(f(n))~~~\text{and}~~~f(2n+1)=f(2n)+1$$ [b](a)[/b] If $f(0)=0$ , find $f(n)$ for every $n$. [b](b)[/b] Show that $f(0)$ cannot equal $1$. [b](c)[/b] For what non-negative integers $k$ (if any) can $f(0)$ equal $2^k$ ?

Shrek throws $5$ balls into $5$ empty bins, where each ball’s target is chosen uniformly at random. After Shrek throws the balls, the probability that there is exactly one empty bin can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

An $n\times n$ square table is divided into $n^2$ unit cells. Some unit segments of the obtained grid (i.e. the side of any unit cell) is colored black so that any unit cell of the given square has exactly one black side. Find [b]a)[/b] the smallest [b]b)[/b] the greatest possible number of black unit segments.

A given convex polyhedron has no vertex which belongs to exactly 3 edges. Prove that the number of faces of the polyhedron that are triangles, is at least 8.

The eyes of a magician are blindfolded while a person $A$ from the audience arranges $n$ identical coins in a row, some are heads and the others are tails. The assistant of the magician asks $A$ to write an integer between $1$ and $n$ inclusive and to show it to the audience. Having seen the number, the assistant chooses a coin and turns it to the other side (so if it was heads it becomes tails and vice versa) and does not touch anything else. Afterwards, the bandages are removed from the magician, he sees the sequence and guesses the written number by $A$. For which $n$ is this possible? [hide=Spoiler hint] The original formulation asks: a) Show that if $n$ is possible, so is $2n$; b) Show that only powers of $2$ are possible; I have omitted this from the above formulation, for the reader's interest. [/hide]

Find all four-digit natural numbers $\overline{xyzw}$ with the property that their sum plus the sum of their digits equals $2003$.

Baron and Peter are playing a game. They are given a simple finite graph $G$ with $n\ge 3$ vertex and $k$ edges that connects the vertices. First Peter labels two vertices A and B, and places a counter at A. Baron starts first. A move for Baron is move the counter along an edge. Peter's move is to remove an edge from the graph. Baron wins if he reaches $B$, otherwise Peter wins. Given the value of $n$, what is the largest $k$ so that Peter can always win?