$10$ cities are connected by one-way air routes in a way so that each city can be reached from any other by several connected flights. Let $n$ be the smallest number of flights needed for a tourist to visit every city and return to the starting city. Clearly $n$ depends on the flight schedule. Find the largest $n$ and the corresponding flight schedule.

Let $S = \{1,..., 999\}$. Determine the smallest integer $m$. for which there exist $m$ two-sided cards $C_1$,..., $C_m$ with the following properties: $\bullet$ Every card $C_i$ has an integer from $S$ on one side and another integer from $S$ on the other side. $\bullet$ For all $x,y \in S$ with $x\ne y$, it is possible to select a card $C_i$ that shows $x$ on one of its sides and another card $C_j$ (with $i \ne j$) that shows $y$ on one of its sides.

Arash and Babak play the following game, taking turns alternatively, on a $1400\times 1401$ table. Arash starts and in his turns he colors $k$, $L$-corners (any three cell of a square). Babak in his turn colors one $2\times 2$ square. Neither player is allowed to recolor any cell. Find all positive integers $k$ for which Arash has a winning strategy.

Let $A$ be the number of ways in which the set $\{ 1, 2, . . . , n\}$ can be partitioned into non-empty subsets. Let $B$ be the number of ways in which the set $\{ 1, 2, . . . , n, n + 1 \}$ can be partitioned into non-empty subsets such that consecutive numbers belong to distinct subsets. Partitions that differ only in the order of the subsets are considered equal. Prove that $A = B$.

Ben and Connor are playing a game of wallball. The first player to lead by $2$ points wins the game. Suppose Ben wins each point with probability $\tfrac{3}{4}$ and is gracious enough to allow Connor to start with a $1$ point lead. The probability that Ben wins the game is $\tfrac{m}{n}$ for coprime positive integers $m$ and $n.$ What is $m + n$?

All cells of $2015 \times 2015$ table colored in one of $4$ colors. We count number of ways to place one tetris T-block in table. Prove that T-block has cell of all $4$ colors in less than $51\%$ of total number of ways.

On a $50 \times 50$ chessboard, we put, in the lower left corner, a die whose faces are numbered from $1$ to $6$. By convention, the sum of digits on two opposite side of the die equals $7$. Adama wants to move the die to the diagonally opposite corner using the following rule: at each step, Adama can roll the die only on to its right side, or to its top side. We suppose that whenever the die lands on a square, the number on its bottom face is printed on the square. By the end of these operations, Adama wants to find the sum of the $99$ numbers appearing on the chessboard. What are the maximum and minimum possible values of this sum?

In the center of each cell of a checkered rectangle $M{}$ there is a point-like light bulb. All the light bulbs are initially switched off. In one turn it is allowed to choose a straight line not intersecting any light bulbs such that on one side of it all the bulbs are switched off, and to switch all of them on. In each turn at least one bulb should be switched on. The task is to switch on all the light bulbs using the largest possible number of turns. What is the maximum number of turns if: [list=a] [*]$M$ is a square of size $21 \times 21$; [*]$M$ is a rectangle of size $20 \times 21$? [/list] [i]Alexandr Shapovalov[/i]

A traveler arrived to an island where 50 natives lived. All the natives stood in a circle and each announced firstly the age of his left neighbour, then the age of his right neighbour. Each native is either a knight who told both numbers correctly or a knave who increased one of the numbers by 1 and decreased the other by 1 (on his choice). Is it always possible after that to establish which of the natives are knights and which are knaves? [i]Alexandr Gribalko[/i]

[b]p1.[/b] The entrance fee the county fair is $64$ cents. Unfortunately, you only have nickels and quarters so you cannot give them exact change. Furthermore, the attendent insists that he is only allowed to change in increments of six cents. What is the least number of coins you will have to pay? [b]p2.[/b] At the county fair, there is a carnival game set up with a mouse and six cups layed out in a circle. The mouse starts at position $A$ and every ten seconds the mouse has equal probability of jumping one cup clockwise or counter-clockwise. After a minute if the mouse has returned to position $A$, you win a giant chunk of cheese. What is the probability of winning the cheese? [b]p3.[/b] A clown stops you and poses a riddle. How many ways can you distribute $21$ identical balls into $3$ different boxes, with at least $4$ balls in the first box and at least $1$ ball in the second box? [b]p4.[/b] Watch out for the pig. How many sets $S$ of positive integers are there such that the product of all the elements of the set is $125970$? [b]p5.[/b] A good word is a word consisting of two letters $A$, $B$ such that there is never a letter $B$ between any two $A$'s. Find the number of good words with length $8$. [b]p6.[/b] Evaluate $\sqrt{2 -\sqrt{2 +\sqrt{2-...}}}$ without looking. [b]p7.[/b] There is nothing wrong with being odd. Of the first $2006$ Fibonacci numbers ($F_1 = 1$, $F_2 = 1$), how many of them are even? [b]p8.[/b] Let $f$ be a function satisfying $f (x) + 2f (27- x) = x$. Find $f (11)$. [b]p9.[/b] Let $A$, $B$, $C$ denote digits in decimal representation. Given that $A$ is prime and $A -B = 4$, nd $(A,B,C)$ such that $AAABBBC$ is a prime. [b]p10.[/b] Given $\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k$ , find $\frac{x^8+y^8}{x^8-y^8}$ in term of $k$. [b]p11.[/b] Let $a_i \in \{-1, 0, 1\}$ for each $i = 1, 2, 3, ..., 2007$. Find the least possible value for $\sum^{2006}_{i=1}\sum^{2007}_{j=i+1} a_ia_j$. [b]p12.[/b] Find all integer solutions $x$ to $x^2 + 615 = 2^n$ for any integer $n \ge 1$. [b]p13.[/b] Suppose a parabola $y = x^2 - ax - 1$ intersects the coordinate axes at three points $A$, $B$, and $C$. The circumcircle of the triangle $ABC$ intersects the $y$ - axis again at point $D = (0, t)$. Find the value of $t$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]6.1 [/b] Two people went from point A to point B. The first one walked along highway at a speed of 5 km/h, and the second along a path at a speed of 4 km/h. The first of them arrived at point B an hour later and traveled 6 kilometers more. Find the distance from A to B along the highway. [b]6.2.[/b] A pedestrian walks along the highway at a speed of 5 km/hour. Along this highway in both directions at the same speed Buses run, meeting every 5 minutes. At 12 o'clock the pedestrian noticed that the buses met near him and, Continuing to walk, he began to count those oncoming and overtaking buses. At 2 p.m., buses met near him again. It turned out that during this time the pedestrian encountered 4 buses more than overtook him. Find the speed of the bus [b]6.3. [/b] Prove that the difference $43^{43} - 17^{17}$ is divisible by $10$. [b]6.4. [/b] Two squares are cut out of the chessboard on the border of the board. When is it possible and when is it not possible to cover with the remaining squares of the board? shapes of the view without overlay? [b]6.5.[/b] The distance from city A to city B (by air) is 30 kilometers, from B to C - 80 kilometers, from C to D - 236 kilometers, from D to E - 86 kilometers, from E to A- 40 kilometers. Find the distance from E to C. [b]6.6.[/b] Is it possible to write down the numbers from $ 1$ to $1963$ in a series so that any two adjacent numbers and any two numbers located one after the other were mutually prime? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here[/url].

In each cell of a $4\times 4$ table, one of the numbers $1$ or $2$ is written. For each row, calculate the sum of the numbers written in it, and for each column, calculate the product of the numbers written in it. Show that some two of the eight results obtained are equal.

Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the sequence $1$, $2$, $\dots$ , $n$ satisfying $$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$. Proposed by United Kingdom

An ant walks on the interior surface of a cube, he moves on a straight line. If ant reaches to an edge the he moves on a straight line on cube's net. Also if he reaches to a vertex he will return his path. a) Prove that for each beginning point ant can has infinitely many choices for his direction that its path becomes periodic. b) Prove that if if the ant starts from point $A$ and its path is periodic, then for each point $B$ if ant starts with this direction, then his path becomes periodic.

The Sultan gathered $300$ court sages and offered them a test. There are caps of $25$ different colors, known in advance to the sages. The Sultan said that one of these caps will be put on each of the sages, and if for each color write the number of caps worn, then all numbers will be different. Every sage can see the caps of the other sages, but not own cap. Then all the sages will simultaneously announce the supposed color of their cap. Can sages advance agree to act in such a way that at least $150$ of them are guaranteed to name a color right?

The points of a circle are colored by three colors. Prove that there exist infinitely many isosceles triangles inscribed in the circle whose vertices are of the same color.

Some towns are connected by roads, with at most one road between any two towns. Let $v$ be the number of towns and $e$ be the number of roads. Prove that $(a)$ if $e<v-1$, then there are two towns such that one cannot travel between them; $(b)$ if $2e>(v-1)(v-2)$, then one can travel between any two towns.

The central square of the City of Mathematicians is an $n\times n$ rectangular shape, each paved with $1\times 1$ tiles. In order to illuminate the square, night lamps are placed at the corners of the tiles (including the edges of the rectangle) in such a way that each night lamp illuminates all the tiles in its corner. Determine the minimum number of night lamps such that even if one of those night lamps does not work, it is possible to illuminate the entire central square with them.

Let $n$ be a positive integer. Marc has $2n$ boxes, and in particular, he has one box filled with $k$ apples for each $k=1,2,3,\ldots,2n$. Every day, Marc opens a box and eats all the apples in it. However, if he eats strictly more than $2n+1$ apples in two consecutive days, he gets stomach ache. Prove that Marc has exactly $2^n$ distinct ways of choosing the boxes so that he eats all the apples but doesn't get stomach ache.

In Sikinia we only pay with coins that have a value of either $11$ or $12$ Kulotnik. In a burglary in one of Sikinia's banks, $11$ bandits cracked the safe and could get away with $5940$ Kulotnik. They tried to split up the money equally - so that everyone gets the same amount - but it just doesn't worked. After a while their leader claimed that it actually isn't possible. Prove that they didn't get any coin with the value $12$ Kulotnik.

In each 1x1 square of a nxn board we write $n^2$ numbers with sum S.A move is choosing a 2x2 square and adding 1 to three numbers(from three different 1x1 squares).We say that a number n is good if we can make all the numbers on the board equal by applying a successive number of moves and it not depends of S. a)Show that 6 is not good b)Show that 4 and 1024 are good

A rectangle is drawn on checkered paper, the sides of which form angles of $45^o$ with the grid lines, and the vertices do not lie on the grid lines. Can an odd number of grid lines intersect each side of a rectangle?

What is the maximal number of crosses than can fit in a $10\times 11$ board without overlapping? Is this problem well-known? [asy] size(4.58cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -3.18, xmax = 1.4, ymin = -0.22, ymax = 3.38; /* image dimensions */ /* draw figures */ draw((-3.,2.)--(1.,2.)); draw((-2.,3.)--(-2.,0.)); draw((-2.,0.)--(-1.,0.)); draw((-1.,0.)--(-1.,3.)); draw((-1.,3.)--(-2.,3.)); draw((-3.,1.)--(1.,1.)); draw((1.,1.)--(1.,2.)); draw((-3.,2.)--(-3.,1.)); draw((0.,2.)--(0.,1.)); draw((-1.,2.)--(-1.,1.)); draw((-2.,2.)--(-2.,1.)); /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]

Let $n\geq 4$ be a positive integer.Out of $n$ people,each of two individuals play table tennis game(every game has a winner).Find the minimum value of $n$,such that for any possible outcome of the game,there always exist an ordered four people group $(a_{1},a_{2},a_{3},a_{4})$,such that the person $a_{i}$ wins against $a_{j}$ for any $1\leq i<j\leq 4$

Consider the set $M=\{1,2,3,...,2003\}$. How many subsets of $M$ with even number of elements exist?