Call a polygon [i]normal[/i] if it can be inscribed in a unit circle. How many non-congruent normal polygons are there such that the square of each side length is a positive integer?

For each positive integer $k,$ let $t(k)$ be the largest odd divisor of $k.$ Determine all positive integers $a$ for which there exists a positive integer $n,$ such that all the differences \[t(n+a)-t(n); t(n+a+1)-t(n+1), \ldots, t(n+2a-1)-t(n+a-1)\] are divisible by 4. [i]Proposed by Gerhard Wöginger, Austria[/i]

At a dinner party there are $N$ hosts and $N$ guests, seated around a circular table, where $N\geq 4$. A pair of two guests will chat with one another if either there is at most one person seated between them or if there are exactly two people between them, at least one of whom is a host. Prove that no matter how the $2N$ people are seated at the dinner party, at least $N$ pairs of guests will chat with one another.

Fatima and Asma are playing the following game. First, Fatima chooses $2013$ pairwise different numbers, called $a_1, a_2, ..., a_{2013}$. Then, Asma tries to know the value of each number $a_1, a_2, ..., a_{2013}$.. At each time, Asma chooses $1 \le i < j \le 2013$ and asks Fatima ''[i]What is the set $\{a_i,a_j\}$?[/i]'' (For example, if Asma asks what is the set $\{a_i,a_j\}$, and $a_1 = 17$ and $a_2 = 13$, Fatima will answer $\{13. 17\}$). Find the least number of questions Asma needs to ask, to know the value of all the numbers $a_1, a_2, ..., a_{2013}$.

Suppose that $1000$ students are standing in a circle. Prove that there exists an integer $k$ with $100 \leq k \leq 300$ such that in this circle there exists a contiguous group of $2k$ students, for which the first half contains the same number of girls as the second half. [i]Proposed by Gerhard Wöginger, Austria[/i]

On a $999\times 999$ board a [i]limp rook[/i] can move in the following way: From any square it can move to any of its adjacent squares, i.e. a square having a common side with it, and every move must be a turn, i.e. the directions of any two consecutive moves must be perpendicular. A [i]non-intersecting route[/i] of the limp rook consists of a sequence of pairwise different squares that the limp rook can visit in that order by an admissible sequence of moves. Such a non-intersecting route is called [i]cyclic[/i], if the limp rook can, after reaching the last square of the route, move directly to the first square of the route and start over. How many squares does the longest possible cyclic, non-intersecting route of a limp rook visit? [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

A table $2 \times 2010$ is divided to unit cells. Ivan and Peter are playing the following game. Ivan starts, and puts horizontal $2 \times 1$ domino that covers exactly two unit table cells. Then Peter puts vertical $1 \times 2$ domino that covers exactly two unit table cells. Then Ivan puts horizontal domino. Then Peter puts vertical domino, etc. The person who cannot put his domino will lose the game. Find who have winning strategy.

On a blackboard, there are $10$ numbers written: $1, 2, \dots, 10$. Nahyun repeatedly performs the following operations. [b](Operation)[/b] Nahyun chooses two numbers from the 10 numbers on the blackboard that are not in a divisor-multiple relationship, erases them, and writes their GCD and LCM on the blackboard. If every two numbers on the blackboard form a divisor-multiple relationship, Nahyun stops the process. What is the maximum number of operations Nahyun can perform? (Note: $a, b$ are in a divisor-multiple relationship iff $a \mid b$ or $b \mid a$.)

Given four weights in geometric progression and an equal arm balance, show how to find the heaviest weight using the balance only twice.

A graph $G=(V,E)$ is given. If at least $n$ colors are required to paints its vertices so that between any two same colored vertices no edge is connected, then call this graph ''$n-$colored''. Prove that for any $n \in \mathbb{N}$, there is a $n-$colored graph without triangles.

In the game of Ric-Rac-Roe, two players take turns coloring squares of a $3 \times 3$ grid in their color; a player wins if they complete a row or column of their color on their turn. If Alice and Bob play this game, picking an uncolored square uniformly at random on their turn, what is the probability that they tie?

For which positive integers $n$ is the following statement true: if $a_1, a_2, ... , a_n$ are positive integers, $a_k \le n$ for all $k$ and $\sum\limits_{k=1}^{{n}}{a_k}=2n$ then it is always possible to choose $a_{i1} , a_{i2} , ..., a_{ij}$ in such a way that the indices $i_1, i_2,... , i_j$ are different numbers, and $\sum\limits_{k=1}^{{{j}}}{a_{ik}}=n$?

$n$ points are given on the surface of a sphere. Show that the surface can be divided into $n$ congruent regions such that each of them contains exactly one of the given points.

Player $A$ and player $B$ play the next game on an $8$ by $8$ square chessboard. They in turn color a field that is not yet colored. One player uses red and the other blue. Player $A$ starts. The winner is the first person to color the four squares of a square of $2$ by $2$ squares with his color somewhere on the board. Prove that player $B$ can always prevent player $A$ from winning.

In the City of Islands there are $2021$ islands connected by bridges. For any given pair of islands $A$ and $B$, one can go from island $A$ to island $B$ using the bridges. Moreover, for any four islands $A_1, A_2, A_3$ and $A_4$: if there is a bridge from $A_i$ to $A_{i+1}$ for each $i \in \left \{ 1,2,3 \right \}$, then there is a bridge between $A_{j}$ and $A_{k}$ for some $j,k \in \left \{ 1,2,3,4 \right \}$ with $|j-k|=2$. Show that there is at least one island which is connected to any other island by a bridge.

Let $n$ be an integer greater than $1.$ $n$ pupils are seated around a round table, each having a certain number of candies (it is possible that some pupils don't have a candy) such that the sum of all the candies they possess is a multiple of $n.$ They exchange their candies as follows: For each student's candies at first, there is at least a student who has more candies than the student sitting to his/her right side, in which case, the student on the right side is given a candy by that student. After a round of exchanging, if there is at least a student who has candies greater than the right side student, then he/she will give a candy to the next student sitting to his/her right side. Prove that after the exchange of candies is completed (ie, when it reaches equilibrium), all students have the same number of candies.

Let $n$ be a positive integer. Elfie the Elf travels in $\mathbb{R}^3$. She starts at the origin: $(0,0,0)$. In each turn she can teleport to any point with integer coordinates which lies at distance exactly $\sqrt{n}$ from her current location. However, teleportation is a complicated procedure: Elfie starts off [i]normal[/i] but she turns [i]strange[/i] with her first teleportation. Next time she teleports she turns [i]normal[/i] again, then [i]strange [/i]again... etc. For which $n$ can Elfie travel to any point with integer coordinates and be [i]normal [/i]when she gets there?

In $n$ transparent boxes there are red balls and blue balls. One needs to choose $50$ boxes such that, together, they contain at least half of the red balls and at least half of the blue balls. Is such a choice possible irrespective on the number of balls and on the way they are distributed in the boxes, if: a) $n = 100$ b) $n = 99$?

Is it possible to divide a $7\times7$ table into a few $\text{connected}$ parts of cells with the same perimeter? ( A group of cells is called $\text{connected}$ if any cell in the group, can reach other cells by passing through the sides of cells.)

A TV station holds a math talent competition, where each participant will be scored by 8 people. The scores are F (failed), G (good), or E (exceptional). The competition is participated by three people, A, B, and C. In the competition, A and B get the same score from exactly 4 people. C states that he has differing scores with A from at least 4 people, and also differing scores with B from at least 4 people. Assuming C tells the truth, how many scoring schemes can occur?

There are $K$ boys placed around a circle. Each of them has an even number of sweets. At a command each boy gives half of his sweets to the boy on his right. If, after that, any boy has an odd number of sweets, someone outside the circle gives him one more sweet to make the number even. This procedure can be repeated indefinitely. Prove that there will be a time at which all boys will have the same number of sweets. (A Andjans, Riga)

Consider the real numbers $a_1,a_2,...,a_{2n}$ whose sum is equal to $0$. Prove that among pairs $(a_i,a_j) , i<j$ where $ i,j \in \{1,2,...,2n\} $ .there are at least $2n-1$ pairs with the property that $a_i+a_j\ge 0$.

Let $A{}$ be a point in the Cartesian plane. At each step, Ann tells Bob a number $0< a\leqslant 1$ and he then moves $A{}$ in one of the four cardinal directions, at his choice, by a distance of $a{}$. This process cotinues as long as Ann wishes. Amongst every $100$ consecutive moves, each of the four possible moves should have been made at least once. Ann's goal is to force Bob to eventually choose a point at a distance greater than $100$ from the initial position of $A{}$. Can Ann achieve her goal?

Alice and Bob determine a number with $2018$ digits in the decimal system by choosing digits from left to right. Alice starts and then they each choose a digit in turn. They have to observe the rule that each digit must differ from the previously chosen digit modulo $3$. Since Bob will make the last move, he bets that he can make sure that the final number is divisible by $3$. Can Alice avoid that? [i](Proposed by Richard Henner)[/i]

Let $\omega=e^{2\pi i/20}$ and let $S$ be the set $\{1, \omega, \ldots, \omega^{19}\}.$ How many subsets of $S$ sum to $0$? Include both $S$ and the empty set in your count.