Let $1\le k\le n$ be integers. At most how many $k$-element subsets can we select from $\{1,2,\dots,n\}$ such that for any two selected subsets, one of the subsets consists of the $k$ smallest elements of their union?

Let $P$ be a finite set of points in the plane. Assume that the distance between any two points is an integer. Prove that $P$ can be colored by three colors such that the distance between any two points of the same color is an even number.

Is it possible to partition the set of integers into three disjoint sets so that for every positive integer $n$ the numbers $n$, $n-50$ and $n+1987$ belong to different sets?

Members of "Professionous Riddlous" society have been divided into some groups, and groups are changed in a special way each weekend: In each group, one of the members is specified as the best member, and the best members of all groups separate from their previous group and form a new group. If a group has only one member, that member joins the new group and the previous group will be removed. Suppose that the society has $n$ members at first, and all the members are in one group. Prove that a week will come, after which number of members of each group will be at most $1+\sqrt{2n}$.

A square is dissected into rectangles with sides parallel to the sides of the square. For each of these rectangles, the ratio of its shorter side to its longer side is considered. Show that the sum of all these ratios is at least $1$.

There are $100$ boxes, each containing either a red cube or a blue cube. Alex has a sum of money initially, and places bets on the colour of the cube in each box in turn. The bet can be anywhere from $0$ up to everything he has at the time. After the bet has been placed, the box is opened. If Alex loses, his bet will be taken away. If he wins, he will get his bet back, plus a sum equal to the bet. Then he moves onto the next box, until he has bet on the last one, or until he runs out of money. What is the maximum factor by which he can guarantee to increase his amount of money, if he knows that the exact number of blue cubes is [list][b](a)[/b] $1$; [b](b)[/b] some integer $k$, $1 < k \leq 100$.[/list]

We start with any finite list of distinct positive integers. We may replace any pair $n, n + 1$ (not necessarily adjacent in the list) by the single integer $n-2$, now allowing negatives and repeats in the list. We may also replace any pair $n, n + 4$ by $n - 1$. We may repeat these operations as many times as we wish. Either determine the most negative integer which can appear in a list, or prove that there is no such minimum.

An $ n \times n$ matrix whose entries come from the set $ S \equal{} \{1, 2, \ldots , 2n \minus{} 1\}$ is called a [i]silver matrix[/i] if, for each $ i \equal{} 1, 2, \ldots , n$, the $ i$-th row and the $ i$-th column together contain all elements of $ S$. Show that: (a) there is no silver matrix for $ n \equal{} 1997$; (b) silver matrices exist for infinitely many values of $ n$.

Yura put $2001$ coins of $1$, $2$ or $3$ kopeykas in a row. It turned out that between any two $1$-kopeyka coins there is at least one coin; between any two $2$-kopeykas coins there are at least two coins; and between any two $3$-kopeykas coins there are at least $3$ coins. How many $3$-koyepkas coins could Yura put?

A square board is divided by lines parallel to the board sides ($7$ lines in each direction, not necessarily equidistant ) into $64$ rectangles. Rectangles are colored into white and black in alternating order. Assume that for any pair of white and black rectangles the ratio between area of white rectangle and area of black rectangle does not exceed $2.$ Determine the maximal ratio between area of white and black part of the board. White (black) part of the board is the total sum of area of all white (black) rectangles.

Fedir and Mykhailo have three piles of stones: the first contains $100$ stones, the second $101$, the third $102$. They are playing a game, going in turns, Fedir makes the first move. In one move player can select any two piles of stones, let's say they have $a$ and $b$ stones left correspondently, and remove $gcd(a, b)$ stones from each of them. The player after whose move some pile becomes empty for the first time wins. Who has a winning strategy? As a reminder, $gcd(a, b)$ denotes the greatest common divisor of $a, b$. [i](Proposed by Oleksii Masalitin)[/i]

In a group of $n$ persons, (i) each person is acquainted to exactly $k$ others, (ii) any two acquainted persons have exactly $l$ common acquaintances, (iii) any two non-acquainted persons have exactly $m$ common acquaintances. Prove that $m(n-k -1) = k(k -l -1)$.

On a board $4\times 4$ the numbers from $1$ to $16$ are written, one in each box. Andres and Pablo choose four numbers each. Andrés chooses the biggest of each row and Pablo, the biggest of each column. The same number can be chosen by both. Then they are removed from the board all chosen numbers. What is the greatest value that the sum of the numbers can have what are left on the board?

In a given community of people, each person has at least two friends within the community. Whenever some people from this community sit on a round table such that each adjacent pair of people are friends, it happens that no non-adjacent pair of people are friends. Prove that there exist two people in this community such that each has exactly two friends and they have at least one common friend.

Let $C$ be a (probably infinite) family of subsets of $\mathbb{N}$ such that for every chain $C_{1}\subset C_{2}\subset \ldots$ of members of $C$, there is a member of $C$ containing all of them. Show that there is a member of $C$ such that no other member of $C$ contains it!

A road has constant width. It is made up of finitely many straight segments joined by corners, where the inner corner is a point and the outer side is a circular arc. The direction of the straight sections is always between $NE$ ($45^o$) and $SSE$ ($157 1/2^o$). A person wishes to walk along the side of the road from point $A$ to point $B$ on the same side. He may only cross the street perpendicularly. What is the shortest route? [figure missing]

A square is divided into $N^2$ equal smaller non-overlapping squares, where $N \geq 3$. We are given a broken line which passes through the centres of all the smaller squares (such a broken line may intersect itself). [list] [*] Show that it is possible to find a broken line composed of $4$ segments for $N = 3$. [*] Find the minimum number of segments in this broken line for arbitrary $N$. [/list]

In each box of a $ 1 \times 2009$ grid, we place either a $ 0$ or a $ 1$, such that the sum of any $ 90$ consecutive boxes is $ 65$. Determine all possible values of the sum of the $ 2009$ boxes in the grid.

Let $ v_1, v_2, \ldots, v_{1989}$ be a set of coplanar vectors with $ |v_r| \leq 1$ for $ 1 \leq r \leq 1989.$ Show that it is possible to find $ \epsilon_r$, $1 \leq r \leq 1989,$ each equal to $ \pm 1,$ such that \[ \left | \sum^{1989}_{r\equal{}1} \epsilon_r v_r \right | \leq \sqrt{3}.\]

Let $n \ge 6$ be an integer. We have at our disposal $n$ colors. We color each of the unit squares of an $n \times n$ board with one of the $n$ colors. a) Prove that, for any such coloring, there exists a path of a chess knight from the bottom-left to the upper-right corner, that does not use all the colors. b) Prove that, if we reduce the number of colors to $\lfloor 2n/3 \rfloor + 2$, then the statement from a) is true for infinitely many values of $n$ and it is false also for infinitely many values of $n$

Along a circle are $100$ white points. An integer $k$ is given, where $2 \le k \le 50$. In each move, we choose a block of $k$ adjacent points such that the first and the last are white, and we paint both of them black. For which values of $k$ is it possible for us to paint all $100$ points black after $50$ moves?

The vertices of a prism are colored using two colors, so that each lateral edge has its vertices differently colored. Consider all the segments that join vertices of the prism and are not lateral edges. Prove that the number of such segments with endpoints differently colored is equal to the number of such segments with endpoints of the same color.

Two players play a game. They have $n > 2$ piles containing $n^{10}+1$ stones each. A move consists of removing all the piles but one and dividing the remaining pile into $n$ nonempty piles. The player that cannot move loses. Who has a winning strategy, the player that moves first or his adversary?

Given is a graph $G$ of $n+1$ vertices, which is constructed as follows: initially there is only one vertex $v$, and one a move we can add a vertex and connect it to exactly one among the previous vertices. The vertices have non-negative real weights such that $v$ has weight $0$ and each other vertex has a weight not exceeding the avarage weight of its neighbors, increased by $1$. Prove that no weight can exceed $n^2$.

How many ways are there to completely fill a $3 \times 3$ grid of unit squares with the letters $B, M$, and $T$, assigning exactly one of the three letters to each of the squares, such that no $2$ adjacent unit squares contain the same letter? Two unit squares are adjacent if they share a side.