Seven classmates are comparing their end-of-year grades in $ 12$ subjects. They observe that for any two of them, there is some subject out of the $ 12$ where the two students got different grades. It is possible to choose n subjects out of the $ 12$ such that if the seven students only compare their grades in these $n$ subjects, it will still be true that for any two, there is some subject out of the n where they got different grades. What is the smallest value of $n$ for which such a selection is surely possible? Note: In Hungarian high schools, students receive an integer grade from $ 1$ to $5$ in each subject at the end of the year.

A student wrote down the following sequence of numbers : the first number is 1, the second number is 2, and after that, each number is obtained by adding together all the previous numbers. Determine the 12th number in the sequence.

Spot spotlight located at vertex $B$ of an equilateral triangle $ABC$, illuminates angle $\alpha$. Find all such values of $\alpha$, not exceeding $60^o$, which at any position of the spotlight, when the illuminated corner is entirely located inside the angle $ABC$, from the illuminated and two unlit segments of side $AC$ can be formed into a triangle.

For each subset $S$ of $\mathbb{N}$, with $r_S (n)$ we denote the number of ordered pairs $(a,b)$, $a,b\in S$, $a\neq b$, for which $a+b=n$. Prove that $\mathbb{N}$ can be partitioned into two subsets $A$ and $B$, so that $r_A(n)=r_B(n)$ for $\forall$ $n\in \mathbb{N}$.

There are $2022$ users on a social network called Mathbook, and some of them are Mathbook-friends. (On Mathbook, friendship is always mutual and permanent.) Starting now, Mathbook will only allow a new friendship to be formed between two users if they have [i]at least two[/i] friends in common. What is the minimum number of friendships that must already exist so that every user could eventually become friends with every other user?

Given is a set of $2n$ cards numbered $1,2, \cdots, n$, each number appears twice. The cards are put on a table with the face down. A set of cards is called good if no card appears twice. Baron Munchausen claims that he can specify $80$ sets of $n$ cards, of which at least one is sure to be good. What is the maximal $n$ for which the Baron's words could be true?

A strip consists of $n$ squares which are numerated in their order by integers $1,2,3,..., n$. In the beginning, one square is empty while each remaining square contains one piece. Whenever a square contains a piece and its some neighbouring square contains another piece while the square immediately following the neighbouring square is empty, one may raise the first piece over the second one to the empty square, removing the second piece from the strip. Find all possibilites which square can be initially empty, if it is possible to reach a state where the strip contains only one piece and a) $n = 2008$, b) $n = 2009$.

A $1995 \times 1995$ square board is given. A coloring of the cells of the board is called [i]good [/i] if the cells can be rearranged so as to produce a colored square board that is symmetric with respect to the main diagonal. Find all values of $k$ for which any $k$-coloring of the given board is [i]good[/i].

In an election, there are a total of $12$ candidates. An election committee has $6$ members voting. It is known that at most two candidates voted by any two committee members are the same. Find the maximum number of committee members.

$m\times n$ grid is tiled by mosaics $2\times2$ and $1\times3$ (horizontal and vertical). Prove that the number of ways to choose a $1\times2$ rectangle (horizontal and vertical) such that one of its cells is tiled by $2\times2$ mosaic and the other cell is tiled by $1\times3$ mosaic [horizontal and vertical] is an even number.

A set $S$ is called [i]neighbouring [/i] if it has the following two properties: a) $S$ has exactly four elements b) for every element $x$ of $S$, at least one of the numbers $x - 1$ or $x+1$ belongs to $S$. Find the number of all [i]neighbouring [/i] subsets of the set $\{1,2,... ,n\}$.

Given a natural number $n$, find the number of quadruples $(x,y,u,v)$ of integers with $1\le x,y,y,v\le n$ satisfy the following inequalities: \begin{align*} &1\le v+x-y\le n,\\ &1\le x+y-u\le n,\\ &1\le u+v-y\le n,\\ &1\le v+x-u\le n. \end{align*}

A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?

World Cup in America introduced a new point system. For a victory $3$ points are given, for a draw $1$ point and for defeat $0$ points. In the preliminary games, the teams are divided into groups of $4$ teams. In groups, teams play with each other, once, then in accordance with the points scored $a,b,c$ and $d$ ($a>b>c>d$) teams take the first, second, third and fourth place in their groups. Give all possible options for the distribution points $a,b,c$ and $d$

Is it possible to cover a plane with circles in such a way that exactly $1988$ circles pass through each point? ( N . Vasiliev)

Armadillo and Badger are playing a game. Armadillo chooses a nonempty tree (a simple acyclic graph) and places apples at some of its vertices (there may be several apples on a single vertex). First, Badger picks a vertex $v_0$ and eats all its apples. Next, Armadillo and Badger take turns alternatingly, with Armadillo starting. On the $i$-th turn the animal whose turn it is picks a vertex $v_i$ adjacent to $v_{i-1}$ that hasn't been picked before and eats all its apples. The game ends when there is no such vertex $v_i$. Armadillo's goal is to have eaten more apples than Badger once the game ends. Can she fulfill her wish?

Let $n=2k-1$, where $k\ge 6$ is an integer. Let $T$ be the set of all $n$-tuples $$\textbf{x}=(x_1,x_2,\dots ,x_n), \text{ where, for } i=1,2,\dots ,n, \text{ } x_i \text{ is } 0 \text{ or } 1.$$ For $\textbf{x}=(x_1,x_2,\dots ,x_n)$ and $\textbf{y}=(y_1,y_2,\dots ,y_n)$ in $T$, let $d(\textbf{x},\textbf{y})$ denote the number of integers $j$ with $1\le j\le n$ such that $x_j\neq x_y$. $($In particular, $d(\textbf{x},\textbf{x})=0)$. Suppose that there exists a subset $S$ of $T$ with $2^k$ elements which has the following property: given any element $\textbf{x}$ in $T$, there is a unique $\textbf{y}$ in $S$ with $d(\textbf{x},\textbf{y})\le 3$. Prove that $n=23$.

How many ways are there to arrange the numbers $1, 2, 3, .. , 15$ in some order such that for any two numbers which are $2$ or $3$ positions apart, the one on the left is greater?

Let $m,n \geqslant 2$ be integers, let $X$ be a set with $n$ elements, and let $X_1,X_2,\ldots,X_m$ be pairwise distinct non-empty, not necessary disjoint subset of $X$. A function $f \colon X \to \{1,2,\ldots,n+1\}$ is called [i]nice[/i] if there exists an index $k$ such that \[\sum_{x \in X_k} f(x)>\sum_{x \in X_i} f(x) \quad \text{for all } i \ne k.\] Prove that the number of nice functions is at least $n^n$.

There are $n$ children around a round table. Erika is the oldest among them and she has $n$ candies, while no other child has any candy. Erika decided to distribute the candies according to the following rules. In every round, she chooses a child with at least two candies and the chosen child sends a candy to each of his/her two neighbors. (So in the first round Erika must choose herself). For which $n \ge 3$ is it possible to end the distribution after a finite number of rounds with every child having exactly one candy?

Let $N$ be a positive integer. A collection of $4N^2$ unit tiles with two segments drawn on them as shown is assembled into a $2N\times2N$ board. Tiles can be rotated. [asy]size(1.5cm);draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);draw((0,0.5)--(0.5,0),red);draw((0.5,1)--(1,0.5),red);[/asy] The segments on the tiles define paths on the board. Determine the least possible number and the largest possible number of such paths. [i]For example, there are $9$ paths on the $4\times4$ board shown below.[/i] [asy]size(4cm);draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);draw((0,1)--(4,1));draw((0,2)--(4,2));draw((0,3)--(4,3));draw((1,0)--(1,4));draw((2,0)--(2,4));draw((3,0)--(3,4));draw((0,3.5)--(0.5,4),red);draw((0,2.5)--(1.5,4),red);draw((3.5,0)--(4,0.5),red);draw((2.5,0)--(4,1.5),red);draw((0.5,0)--(0,0.5),red);draw((2.5,4)--(3,3.5)--(3.5,4),red);draw((4,3.5)--(3.5,3)--(4,2.5),red);draw((0,1.5)--(1,2.5)--(1.5,2)--(0.5,1)--(1.5,0),red);draw((1.5,3)--(2,3.5)--(3.5,2)--(2,0.5)--(1.5,1)--(2.5,2)--cycle,red);[/asy]

A square is divided into rectangles. A "chain" is a subset $K$ of the set of these rectangles such that there exists a side of the square which is covered by projections of rectangles of $K$ and such that no point of this side is a projection of two inner points of two inner points of two different rectangles of $K$. (a) Prove that every two rectangles in such a division are members of a certain "chain". (b) Solve the similar problem for a cube, divided into rectangular parallelopipeds (in the definition of chain , replace "side" by"edge") . (A.I . Golberg, V.A. Gurevich)

We consider an integer $n \ge 3,$ the set $S=\{1,2,3,\ldots,n\}$ and the set $\mathcal{F}$ of the functions from $S$ to $S.$ We say that $\mathcal{G} \subset \mathcal{F}$ is a generating set for $\mathcal{H} \subset \mathcal{F}$ if any function in $\mathcal{H}$ can be represented as a composition of functions from $\mathcal{G}.$ a) Let the functions $a:S \to S,$ $a(n-1)=n,$ $a(n)=n-1$ and $a(k)=k$ for $k \in S \setminus \{n-1,n\}$ and $b:S \to S,$ $b(n)=1$ and $b(k)=k+1$ for $k \in S \setminus \{n\}.$ Prove that $\{a,b\}$ is a generating set for the set $\mathcal{B}$ of bijective functions of $\mathcal{F}.$ b) Prove that the smallest number of elements that a generating set of $\mathcal{F}$ has is $3.$

We play a game of musical chairs with $n$ chairs numbered $1$ to $n$. You attach $n$ leaves, numbered $1$ to $n$, to the chairs in such a way that the number on a leaf does not match the number on the chair it is attached to. One player sits on each chair. Every time you clap, each player looks at the number on the leaf attached to his current seat and moves to sit on the seat with that number. Prove that, for any $m$ that is not a prime power with$ 1 < m \leq n$, it is possible to attach the leaves to the seats in such a way that after $m$ claps everyone has returned to the chair they started on for the first time.