(a) Can it happen that in a group of $10$ girls and $9$ boys, ball the girls know a different number of boys while all the boys know the same number of girls? (b) What if there are $11$ girls and $10$ boys? (NB Vassiliev)

Let $n$ be a positive integer and $a_1,a_2,\ldots a_{2n+1}$ be positive reals. For $k=1,2,\ldots ,2n+1$, denote $b_k = \max_{0\le m\le n}\left(\frac{1}{2m+1} \sum_{i=k-m}^{k+m} a_i \right)$, where indices are taken modulo $2n+1$. Prove that the number of indices $k$ satisfying $b_k\ge 1$ does not exceed $2\sum_{i=1}^{2n+1} a_i$.

Let $S$ be a set with 2002 elements, and let $N$ be an integer with $0 \leq N \leq 2^{2002}$. Prove that it is possible to color every subset of $S$ either black or white so that the following conditions hold: (a) the union of any two white subsets is white; (b) the union of any two black subsets is black; (c) there are exactly $N$ white subsets.

The image shown below is a cross with length 2. If length of a cross of length $k$ it is called a $k$-cross. (Each $k$-cross ahs $6k+1$ squares.) [img]http://aycu08.webshots.com/image/4127/2003057947601864020_th.jpg[/img] a) Prove that space can be tiled with $1$-crosses. b) Prove that space can be tiled with $2$-crosses. c) Prove that for $k\geq5$ space can not be tiled with $k$-crosses.

Nair and Yuli play the following game: $1.$ There is a coin to be moved along a horizontal array with $203$ cells. $2.$ At the beginning, the coin is at the first cell, counting from left to right. $3.$ Nair plays first. $4.$ Each of the players, in their turns, can move the coin $1$, $2$, or $3$ cells to the right. $5.$ The winner is the one who reaches the last cell first. What strategy does Nair need to use in order to always win the game?

A square is divided into congruent rectangles with sides of integer lengths. A rectangle is important if it has at least one point in common with a given diagonal of the square. Prove that this diagonal bisects the total area of the important rectangles

A strip of width $w$ is the set of all points which lie on, or between, two parallel lines distance $w$ apart. Let $S$ be a set of $n$ ($n \ge 3$) points on the plane such that any three different points of $S$ can be covered by a strip of width $1$. Prove that $S$ can be covered by a strip of width $2$.

A city is a point on the plane. Suppose there are $n\geq 2$ cities. Suppose that for each city $X$, there is another city $N(X)$ that is strictly closer to $X$ than all the other cities. The government builds a road connecting each city $X$ and its $N(X)$; no other roads have been built. Suppose we know that, starting from any city, we can reach any other city through a series of road. We call a city $Y$ [i]suburban[/i] if it is $N(X)$ for some city $X$. Show that there are at least $(n-2)/4$ suburban cities. [i]Proposed by usjl.[/i]

At each vertex of a regular $14$-gon, lies a coin. Initially $7$ coins are heads, and $7$ coins are tails. Determine the minimum number $t$ such that it’s always possible to turn over at most $t$ of the coins so that in the resulting $14$-gon, no two adjacent coins are both heads and no two adjacent coins are both tails.

Let $p = (a_{1}, a_{2}, \ldots , a_{12})$ be a permutation of $1, 2, \ldots, 12$. We will denote \[S_{p} = |a_{1}-a_{2}|+|a_{2}-a_{3}|+\ldots+|a_{11}-a_{12}|\]We'll call $p$ $\textit{optimistic}$ if $a_{i} > \min(a_{i-1}, a_{i+1})$ $\forall i = 2, \ldots, 11$. $a)$ What is the maximum possible value of $S_{p}$. How many permutations $p$ achieve this maximum?$\newline$ $b)$ What is the number of $\textit{optimistic}$ permtations $p$? $c)$ What is the maximum possible value of $S_{p}$ for an $\textit{optimistic}$ $p$? How many $\textit{optimistic}$ permutations $p$ achieve this maximum?

Let $k$ be a positive integer. There are $2k$ pieces arranged in a row. A [i]move[/i] consists of swapping two adjacent pieces. Several moves must be made so that each piece passes through both the first and the last position. What is the minimum number of moves required to achieve this?

A $k\times \ell$ 'parallelogram' is drawn on a paper with hexagonal cells (it consists of $k$ horizontal rows of $\ell$ cells each). In this parallelogram a set of non-intersecting sides of hexagons is chosen; it divides all the vertices into pairs. Juniors) How many vertical sides can there be in this set? Seniors) How many ways are there to do that? [asy] size(120); defaultpen(linewidth(0.8)); path hex = dir(30)--dir(90)--dir(150)--dir(210)--dir(270)--dir(330)--cycle; for(int i=0;i<=3;i=i+1) { for(int j=0;j<=2;j=j+1) { real shiftx=j*sqrt(3)/2+i*sqrt(3),shifty=j*3/2; draw(shift(shiftx,shifty)*hex); } } [/asy] [i](T. Doslic)[/i]

Let $n$ be a natural number larger than $2$. Vanessa has $n$ piles of jade stones, and all the piles have a different number of stones. Vanessa can distribute the stones from any pile by the other piles and stay with $n-1$ piles with the same number of stones. She also can distribute the stones from any two piles by the other piles and stay with $n-2$ piles with the same number of stones. Find the smallest possible number of jade's stones that the pile with the largest number of stones can have.

Let $n \in \mathbb{N}^*$ and consider a circle of length $6n$ along with $3n$ points on the circle which divide it into $3n$ arcs: $n$ of them have length $1,$ some other $n$ have length $2$ and the remaining $n$ have length $3.$ Prove that among these points there must be two such that the line that connects them passes through the center of the circle.

Quasi-rounding is a substitution one of the two closest integers instead of the given number. Given $n$ numbers. Prove that you can quasi-round them in such a way, that a sum of every subset of quasi-rounded numbers will deviate from the sum of the same subset of initial numbers not greater than $(n+1)/4$ .

There are $ n\geq 5$ people in a party. Assume that among any three of them some two know each other. Show that one can select at least $ \frac{n}{2}$ people and arrange them at a round table so that each person sits between two of his/her acquaintances.

Let $m$ and $n$ be positive integers. Show that $\frac{(m+n)!}{(m+n)^{m+n}} < \frac{m!}{m^m}\cdot\frac{n!}{n^n}$

[b]7.1[/b] Prove that a natural number with an odd number of divisors is a perfect square. [b]7.2[/b] In a triangle $ABC$ with area $S$, medians $AK$ and $BE$ are drawn, intersecting at the point $O$. Find the area of the quadrilateral $CKOE$. [img]https://cdn.artofproblemsolving.com/attachments/0/f/9cd32bef4f4459dc2f8f736f7cc9ca07e57d05.png[/img] [b]7.3 .[/b] The front tires of a car wear out after $25,000$ kilometers, and the rear tires after $15,000$ kilometers. When you need to swap tires so that the car can travel the longest possible distance with the same tires? [b]7.4 [/b] A $24 \times 60$ rectangle is divided by lines parallel to it sides, into unit squares. How many parts will this rectangle be divided into if you also draw a diagonal in it? [b]7.5 / 8.4[/b] Let $ [A]$ denote the largest integer not greater than $A$. Solve the equation: $[(5 + 6x)/8] = (15x-7)/5$ . [b]7.6[/b] Black paint was sprayed onto a white surface. Prove that there are two points of the same color, the distance between which is $1965$ meters. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988081_1965_leningrad_math_olympiad]here[/url].

Ali has $6$ types of $2\times2$ squares with cells colored in white or black, and has presented them to Mohammad as forbidden tiles. $a)$ Prove that Mohammad can color the cells of the infinite table (from each $4$ sides.) in black or white such that there's no forbidden tiles in the table. $b)$ Prove that Ali can present $7$ forbidden tiles such that Mohammad cannot achieve his goal.

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 $\frac{n+1}4$.

There are n cars waiting at distinct points of a circular race track. At the starting signal each car starts. Each car may choose arbitrarily which of the two possible directions to go. Each car has the same constant speed. Whenever two cars meet they both change direction (but not speed). Show that at some time each car is back at its starting point.

Keith has $10$ coins labeled $1$ through $10$, where the $i$th coin has weight $2^i$. The coins are all fair, so the probability of flipping heads on any of the coins is $\tfrac{1}{2}$. After flipping all of the coins, Keith takes all of the coins which land heads and measures their total weight, $W$. If the probability that $137\le W\le 1061$ is $\tfrac{m}{n}$ for coprime positive integers $m,n$, determine $m+n$.

The game of Penta is played with teams of five players each, and there are five roles the players can play. Each of the five players chooses two of five roles they wish to play. If each player chooses their roles randomly, what is the probability that each role will have exactly two players?

A graph has $17$ points and each point has $4$ edges. Show that there are two points which are not joined and which are not both joined to the same point.

Let $M$ be the set of the integer numbers from the range $[-n, n]$. The subset $P$ of $M$ is called a [i]base subset[/i] if every number from $M$ can be expressed as a sum of some different numbers from $P$. Find the smallest natural number $k$ such that every $k$ numbers that belongs to $M$ form a base subset.