Two magicians show the following trick. The first magician goes out of the room. The second magician takes a deck of $100$ cards labelled by numbers $1,2,\ldots ,100$ and asks three spectators to choose in turn one card each. The second magician sees what card each spectator has taken. Then he adds one more card from the rest of the deck. Spectators shuffle these $4$ cards, call the first magician and give him these $4$ cards. The first magician looks at the $4$ cards and “guesses” what card was chosen by the first spectator, what card by the second and what card by the third. Prove that the magicians can perform this trick.

Let $n \ge 2$ be an integer. Consider an $n \times n$ chessboard consisting of $n^2$ unit squares. A configuration of $n$ rooks on this board is [i]peaceful[/i] if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that, for each peaceful configuration of $n$ rooks, there is a $k \times k$ square which does not contain a rook on any of its $k^2$ unit squares.

We want to colour all the squares of an $ nxn$ board of red or black. The colorations should be such that any subsquare of $ 2x2$ of the board have exactly two squares of each color. If $ n\geq 2$ how many such colorations are possible?

We have $n$ points in the plane, no three on a line. We call $k$ of them good if they form a convex polygon and there is no other point in the convex polygon. Suppose that for a fixed $k$ the number of $k$ good points is $c_k$. Show that the following sum is independent of the structure of points and only depends on $n$ : \[ \sum_{i=3}^n (-1)^i c_i \]

A row of 2021 balls is given. Pasha and Vova play a game, taking turns to perform moves; Pasha begins. On each turn a boy should paint a non-painted ball in one of the three available colors: red, yellow, or green (initially all balls are non-painted). When all the balls are colored, Pasha wins, if there are three consecutive balls of different colors; otherwise Vova wins. Who has a winning strategy?

Fix positive integers $n,j,k$.How many integer sequences are there of the form $1\le a_1<a_2<\ldots<a_k\le n$,where $a_{i+1}-a_i\ge j$ for all $1\le i\le k-1$.

Given natural numbers $a$ and $b$, such that $a<b<2a$. Some cells on a graph are colored such that in every rectangle with dimensions $A \times B$ or $B \times A$, at least one cell is colored. For which greatest $\alpha$ can you say that for every natural number $N$ you can find a square $N \times N$ in which at least $\alpha \cdot N^2$ cells are colored?

A $(2n + 1) \times (2n + 1)$ grid, with $n> 0$, is colored in such a way that each of the cell is white or black. A cell is called [i]special[/i] if there are at least $n$ other cells of the same color in its row, and at least another $n$ cells of the same color in its column. (a) Prove that there are at least $2n + 1$ special boxes. (b) Provide an example where there are at most $4n$ special cells. (c) Determine, as a function of $n$, the minimum possible number of special cells.

With $21$ pieces, some white and some black, a rectangle is formed of $3 \times 7$. Prove that there are always four pieces of the same color located at the vertices of a rectangle.

In a $2\times n$ array we have positive reals s.t. the sum of the numbers in each of the $n$ columns is $1$. Show that we can select a number in each column s.t. the sum of the selected numbers in each row is at most $\frac{n+1}4$.

Let $m$ and $n$ be positive integers. A sequence of points $(A_0,A_1,\ldots,A_n)$ on the Cartesian plane is called [i]interesting[/i] if $A_i$ are all lattice points, the slopes of $OA_0,OA_1,\cdots,OA_n$ are strictly increasing ($O$ is the origin) and the area of triangle $OA_iA_{i+1}$ is equal to $\frac{1}{2}$ for $i=0,1,\ldots,n-1$. Let $(B_0,B_1,\cdots,B_n)$ be a sequence of points. We may insert a point $B$ between $B_i$ and $B_{i+1}$ if $\overrightarrow{OB}=\overrightarrow{OB_i}+\overrightarrow{OB_{i+1}}$, and the resulting sequence $(B_0,B_1,\ldots,B_i,B,B_{i+1},\ldots,B_n)$ is called an [i]extension[/i] of the original sequence. Given two [i]interesting[/i] sequences $(C_0,C_1,\ldots,C_n)$ and $(D_0,D_1,\ldots,D_m)$, prove that if $C_0=D_0$ and $C_n=D_m$, then we may perform finitely many [i]extensions[/i] on each sequence until the resulting two sequences become identical.

One wants to distribute $n$ same sized cakes between $k$ people equally by cutting every cake at most once. If the number of positive divisors of $n$ is denoted as $d(n)$, show that the number of different values of $k$ which makes such distribution possible is $n+d(n)$

1. Around a circumference are written $2012$ number, each of with is equal to $1$ or $-1$. If there are not $10$ consecutive numbers that sum $0$, find all possible values of the sum of the $2012$ numbers.

In a row there are $12$ cards that can be of three kinds: with both white faces, with both black faces or with one white face and the other black. Initially there are $9$ cards with the black side facing up. The first six cards from the left are turned over, leaving $9$ cards with the black face up. The six cards on the left are then turned over, leaving $8$ cards with the black face up. Finally, six cards are turned over: the first three on the left and the last three on the right, leaving $3$ cards with the black face up. Decide if with this information it is possible to know with certainty how many cards of each kind are in the row

For any non-empty subset $A$ of $\{1, 2, \ldots , n\}$ define $f(A)$ as the largest element of $A$ minus the smallest element of $A$. Find $\sum f(A)$ where the sum is taken over all non-empty subsets of $\{1, 2, \ldots , n\}$.

From three boys and three girls, every boy knows exactly two girls and every girl knows exactly two boys. Prove that we can arrange boys and girls in pairs such that in every pair people know each other

There is a single coin on each square of a $5 \times 5$ board. All the coins look the same. Two of them are fakes and have equal weight. Genuine coins are heavier than fake ones and also weigh the same. The fake coins are on the squares sharing just one vertice. Is it possible to determine for sure a) 13 genuine coins; b) 15 genuine coins; and c) 17 genuine coins in a single weighing on a balance with no unit weights? [i]Rustem Zhenodarov, Alexandr Gribalko, Sergey Tokarev[/i]

In a $7\times 7\times 7$ cube, the unit cubes are colored white, black and gray colors so that for any two colors the number of cubes of these two colors are different. In this case, $N$ parallel rows of $7$ cubes were found, each of which there are more white cubes than gray and than black. Likewise, there were $N$ parallel rows of $7$ cubes, each of which contained gray there are more cubes than white and than black, and there are also N parallel rows of $7$ cubes, each of which contains more black cubes than white ones and than gray ones. What is the largest $N$ for which this is possible?

Let $ A \equal{} \{(a_1,\dots,a_8)|a_i\in\mathbb{N}$ , $ 1\leq a_i\leq i \plus{} 1$ for each $ i \equal{} 1,2\dots,8\}$.A subset $ X\subset A$ is called sparse if for each two distinct elements $ (a_1,\dots,a_8)$,$ (b_1,\dots,b_8)\in X$,there exist at least three indices $ i$,such that $ a_i\neq b_i$. Find the maximal possible number of elements in a sparse subset of set $ A$.

Find the maximal possible value of $n$ such that there exist points $P_1,P_2,P_3,\ldots,P_n$ in the plane and real numbers $r_1,r_2,\ldots,r_n$ such that the distance between any two different points $P_i$ and $P_j$ is $r_i+r_j$.

A strip of width $1$ is to be divided by rectangular panels of common width $1$ and denominations long $a_1$, $a_2$, $a_3$, $. . .$ be paved without gaps ($a_1 \ne 1$). From the second panel on, each panel is similar but not congruent to the already paved part of the strip. When the first $n$ slabs are laid, the length of the paved part of the strip is $sn$. Given $a_1$, is there a number that is not surpassed by any $s_n$? The accuracy answer has to be proven.

Paint the grid points of $L=\{0,1,\dots,n\}^2$ with red or green in such a way that every unit lattice square in $L$ has exactly two red vertices. How many such colorings are possible?

A convex solid is formed in four-dimensional Euclidean space with vertices at the $24$ possible permutations of $\{1, 2, 3, 4\}$ (so $(1, 2, 3, 4)$, $(1, 2, 4, 3)$, etc.). What is the product of the number of faces and edges of this solid?

Consider a tree with $n$ vertices, labeled with $1,\ldots,n$ in a way that no label is used twice. We change the labeling in the following way - each time we pick an edge that hasn't been picked before and swap the labels of its endpoints. After performing this action $n-1$ times, we get another tree with its labeling a permutation of the first graph's labeling. Prove that this permutation contains exactly one cycle.

A chessboard with some unit squares marked is called a $\textit{good board}$ if for any pair of rows \((s, t)\), a rook placed on a marked square in row \(s\) can reach a marked square in row \(t\) in several moves by only moving to marked squares above, below, or to the right of its current position. Consider a chessboard with 220 rows and 12 columns, where exactly 9 unit squares in each row are marked. Regardless of how the marked squares are chosen, if it is possible to delete \(k\) columns and rearrange the remaining columns to form a $\textit{good board}$ determine the maximum possible value of \(k\).