There are $1999$ people participating in an exhibition. Among any $50$ people there are two who don't know each other. Prove that there are $41$ people, each of whom knows at most $1958$ people.

Let $A = (a_1, a_2, \ldots, a_{2001})$ be a sequence of positive integers. Let $m$ be the number of 3-element subsequences $(a_i,a_j,a_k)$ with $1 \leq i < j < k \leq 2001$, such that $a_j = a_i + 1$ and $a_k = a_j + 1$. Considering all such sequences $A$, find the greatest value of $m$.

Lisa writes a positive whole number in the decimal system on the blackboard and now makes in each turn the following: The last digit is deleted from the number on the board and then the remaining shorter number (or 0 if the number was one digit) becomes four times the number deleted number added. The number on the board is now replaced by the result of this calculation. Lisa repeats this until she gets a number for the first time was on the board. (a) Show that the sequence of moves always ends. (b) If Lisa begins with the number $53^{2022} - 1$, what is the last number on the board? Example: If Lisa starts with the number $2022$, she gets $202 + 4\cdot 2 = 210$ in the first move and overall the result $$2022 \to 210 \to 21 \to 6 \to 24 \to 18 \to 33 \to 15 \to 21$$. Since Lisa gets $21$ for the second time, the turn order ends. [i](Stephan Pfannerer)[/i]

For any positive integer $n$, define $$c_n=\min_{(z_1,z_2,...,z_n)\in\{-1,1\}^n} |z_1\cdot 1^{2018} + z_2\cdot 2^{2018} + ... + z_n\cdot n^{2018}|.$$ Is the sequence $(c_n)_{n\in\mathbb{Z}^+}$ bounded?

Let $n $ be a positive integer. Let $f $ be a function from nonnegative integers to themselves. Let $f (0,i)=f (i,0)=0$, $f (1, 1)=n $, and $ f(i, j)= [\frac {f(i-1,j)}{2}]+ [\frac {f(i, j-1)}{2}] $ for positive integers $i, j$ such that $i*j>1$. Find the number of pairs $(i,j) $ such that $f (i, j) $ is an odd number.( $[x]$ is the floor function).

City of Mar del Plata is a square shaped $WSEN$ land with $2(n + 1)$ streets that divides it into $n \times n$ blocks, where $n$ is an even number (the leading streets form the perimeter of the square). Each block has a dimension of $100 \times 100$ meters. All streets in Mar del Plata are one-way. The streets which are parallel and adjacent to each other are directed in opposite direction. Street $WS$ is driven in the direction from $W$ to $S$ and the street $WN$ travels from $W$ to $N$. A street cleaning car starts from point $W$. The driver wants to go to the point $E$ and in doing so, he must cross as much as possible roads. What is the length of the longest route he can go, if any $100$-meter stretch cannot be crossed more than once? (The figure shows a plan of the city for $n=6$ and one of the possible - but not the longest - routes of the street cleaning car. See http://goo.gl/maps/JAzD too.) [img]http://s14.postimg.org/avfg7ygb5/CPS_2012_P5.jpg[/img]

Ana, Beto, Carlos, Diana, Elena and Fabian are in a circle, located in that order. Ana, Beto, Carlos, Diana, Elena and Fabian each have a piece of paper, where are written the real numbers $a,b,c,d,e,f$ respectively. At the end of each minute, all the people simultaneously replace the number on their paper by the sum of three numbers; the number that was at the beginning of the minute on his paper and on the papers of his two neighbors. At the end of the minute $2022, 2022$ replacements have been made and each person have in his paper it´s initial number. Find all the posible values of $abc+def$. $\textbf{Note:}$ [i]If at the beginning of the minute $N$ Ana, Beto, Carlos have the numbers $x,y,z$, respectively, then at the end of the minute $N$, Beto is going to have the number $x+y+z$[/i].

Given a simple, connected graph with $n$ vertices and $m$ edges. Prove that one can find at least $m$ ways separating the set of vertices into two parts, such that the induced subgraphs on both parts are connected.

Given a table $4\times 4$. a) Find, how $7$ stars can be put in its fields in such a way, that erasing of two arbitrary lines and two columns will always leave at list one of the stars. b) Prove that if there are less than $7$ stars, You can always find two columns and two rows, such, that if you erase them, no star will remain in the table.

We are given a bag of sugar, a two-pan balance, and a weight of $1$ gram. How do we obtain $1$ kilogram of sugar in the smallest possible number of weighings?

What is the largest possible value of $|a_1 - 1| + |a_2-2|+...+ |a_n- n|$ where $a_1, a_2,..., a_n$ is a permutation of $1,2,..., n$?

Given is a chessboard 8x8. We have to place $n$ black queens and $n$ white queens, so that no two queens attack. Find the maximal possible $n$. (Two queens attack each other when they have different colors. The queens of the same color don't attack each other)

a) There is an infinite sequence of $0,1$, like $\dots,a_{-1},a_{0},a_{1},\dots$ (i.e. an element of $\{0,1\}^{\mathbb Z}$). At each step we make a new sequence. There is a function $f$ such that for each $i$, $\mbox{new }a_{i}=f(a_{i-100},a_{i-99},\dots,a_{i+100})$. This operation is mapping $F: \{0,1\}^{\mathbb Z}\longrightarrow\{0,1\}^{\mathbb Z}$. Prove that if $F$ is 1-1, then it is surjective. b) Is the statement correct if we have an $f_{i}$ for each $i$?

15 of the cells of a chessboard 8x8 are chosen. We draw the segments which unite the centers of every two of the chosen squares. Prove that among these segments there are four segments which have the same length.

On each of the cards written in $2013$ by number, all of these $2013$ numbers are different. The cards are turned down by numbers. In a single move is allowed to point out the ten cards and in return will report one of the numbers written on them (do not know what). For what most $w$ guaranteed to be able to find $w$ cards for which we know what numbers are written on each of them?

Let $k>1$ be a given positive integer. A set $S$ of positive integers is called [i]good[/i] if we can colour the set of positive integers in $k$ colours such that each integer of $S$ cannot be represented as sum of two positive integers of the same colour. Find the greatest $t$ such that the set $S=\{a+1,a+2,\ldots ,a+t\}$ is [i]good[/i] for all positive integers $a$. [i]A. Ivanov, E. Kolev[/i]

[list] [*] A test was conducted in class. It is known that at least $\frac{2}{3}$ of the problems were hard. Each such problems were not solved by at least $\frac{2}{3}$ of the students. It is also known that at least $\frac{2}{3}$ of the students passed the test. Each such student solved at least $\frac{2}{3}$ of the suggested problems. Is this possible? [*] Previous problem with $\frac{2}{3}$ replaced by $\frac{3}{4}$. [*] Previous problem with $\frac{2}{3}$ replaced by $\frac{7}{10}$.[/list]

We choose $n > 3$ points on a circle and number them $1$ to $ n$ in some order. We say that two non-adjacent points $A$ and $B$ are related if, in one of the arcs $AB$, all the points are marked with numbers less than those at $A,B$. Show that the number of pairs of related points is exactly $n-3$.

Let $A$ be a set with $1000$ members and $\mathcal F =${$A_1,A_2,\ldots,A_n$} a family of subsets of A such that (a) Each element in $\mathcal F$ consists of 3 members (b) For every five elements in $\mathcal F$, the union of them all will have at least $12$ members Find the largest value of $n$

A $4\times4$ grid has its 16 cells colored arbitrarily in three colors. A [i]swap[/i] is an exchange between the colors of two cells. Prove or disprove that it always takes at most three swaps to produce a line of symmetry, regardless of the grid's initial coloring. [i]Proposed by Matthew Babbitt[/i]

$18.$ A subset $B$ of the set of first $100$ positive integers has the property that no two elements of $B$ sum to $125.$ What is the maximum possible number of elements in $B ?$

Five points are given in the plane that each of $10$ triangles they define has area greater than $2$. Prove that there exists a triangle of area greater than $3$.

It is given regular polygon with $2n$ sides and center $S$. Consider every quadrilateral with vertices as vertices of polygon. Let $u$ be number of such quadrilaterals which contain point $S$ inside and $v$ number of remaining quadrilaterals. Find $u-v$

$A_1B_1A_2$, $B_1A_2B_2$, $A_2B_2A_3$,...,$B_{13}A_{14}B_{14}$, $A_{14}B_{14}A_1$ and $B_{14}A_1B_1$ are equilateral rigid plates that can be folded along the edges $A_1B_1$,$B_1A_2$, ..., $A_{14}B_{14}$ and $B_{14}A_1$ respectively. Can they be folded so that all $28$ plates lie in the same plane?

The number $N$ consists only $2's$ and $1's$ in its [b]decimal representation[/b].We know that,after deleting digits from N,we can get any number consisting $9999$- $1's$ and $one$ - $2's$ in its [b]decimal representation[/b].[b][u]Find the least number of digits in the decimal representation of N[/u][/b]