Let $f(n)$ be the least number of distinct points in the plane such that for each $k = 1, 2, \cdots, n$ there exists a straight line containing exactly $k$ of these points. Find an explicit expression for $f(n).$ [i]Simplified version.[/i] Show that $f(n)=\left[\frac{n+1}{2}\right]\left[\frac{n+2}{2}\right].$ Where $[x]$ denoting the greatest integer not exceeding $x.$

$N$ unit squares on the infinite sheet of cross-lined paper are painted with black colour. Prove that you can cut out the finite number of square pieces and satisfy two conditions all the black squares are contained in those pieces the area of black squares is not less than $1/5$ and not greater than $4/5$ of every piece area.

A kingdom consists of $12$ cities located on a one-way circular road. A magician comes on the $13$th of every month to cast spells. He starts at the city which was the 5th down the road from the one that he started at during the last month (for example, if the cities are numbered $1–12$ clockwise, and the direction of travel is clockwise, and he started at city #$9$ last month, he will start at city #$2$ this month). At each city that he visits, the magician casts a spell if the city is not already under the spell, and then moves on to the next city. If he arrives at a city which is already under the spell, then he removes the spell from this city, and leaves the kingdom until the next month. Last Thanksgiving the capital city was free of the spell. Prove that it will be free of the spell this Thanksgiving as well.

Spot spotlight located at vertex $B$ of an equilateral triangle $ABC$, illuminates angle $\alpha$. Find all such values of $\alpha$, not exceeding $60^o$, which at any position of the spotlight, when the illuminated corner is entirely located inside the angle $ABC$, from the illuminated and two unlit segments of side $AC$ can be formed into a triangle.

A strip consists of $n$ squares which are numerated in their order by integers $1,2,3,..., n$. In the beginning, one square is empty while each remaining square contains one piece. Whenever a square contains a piece and its some neighbouring square contains another piece while the square immediately following the neighbouring square is empty, one may raise the first piece over the second one to the empty square, removing the second piece from the strip. Find all possibilites which square can be initially empty, if it is possible to reach a state where the strip contains only one piece and a) $n = 2008$, b) $n = 2009$.

A $1995 \times 1995$ square board is given. A coloring of the cells of the board is called [i]good [/i] if the cells can be rearranged so as to produce a colored square board that is symmetric with respect to the main diagonal. Find all values of $k$ for which any $k$-coloring of the given board is [i]good[/i].

Is it possible to cover a plane with circles in such a way that exactly $1988$ circles pass through each point? ( N . Vasiliev)

A square is divided into rectangles. A "chain" is a subset $K$ of the set of these rectangles such that there exists a side of the square which is covered by projections of rectangles of $K$ and such that no point of this side is a projection of two inner points of two inner points of two different rectangles of $K$. (a) Prove that every two rectangles in such a division are members of a certain "chain". (b) Solve the similar problem for a cube, divided into rectangular parallelopipeds (in the definition of chain , replace "side" by"edge") . (A.I . Golberg, V.A. Gurevich)

A positive integer $m(\ge 2$) is given. From circle $C_1$ with a radius 1, construct $C_2, C_3, C_4, ... $ through following acts: In the $i$th act, select a circle $P_i$ inside $C_i$ with a area $\frac{1}{m}$ of $C_i$. If such circle dosen't exist, the act ends. If not, let $C_{i+1}$ a difference of sets $C_i -P_i$. Prove that this act ends within a finite number of times.

Given a billiard in the form of a regular $1998$-gon $A_1A_2...A_{1998}$. A ball was released from the midpoint of side $A_1A_2$, which, reflected therefore from sides $A_2A_3$, $A_3A_4$, . . . , $A_{1998}A_1$ (according to the law, the angle of incidence is equal to the angle of reflection), returned to the starting point. Prove that the trajectory of the ball is a regular $1998$-gon.

Let $n \ge 3$ be an integer, and consider a $n \times n$-board, divided into $n^2$ unit squares. For all $m \ge 1$, arbitrarily many $1\times m$-rectangles (type I) and arbitrarily many $m\times 1$-rectangles (type II) are available. We cover the board with $N$ such rectangles, without overlaps, and such that every rectangle lies entirely inside the board. We require that the number of type I rectangles used is equal to the number of type II rectangles used.(Note that a $1 \times 1$-rectangle has both types.) What is the minimal value of $N$ for which this is possible?

Given $n$ points, $n > 4$. Prove that tou can connect them with arrows, in such a way, that you can reach every point from every other point, having passed through one or two arrows. (You can connect every pair with one arrow only, and move along the arrow in one direction only.)

Given points $O$ and $A$ in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point $X$ in the plane, the circle $C(X)$ has center $O$ and radius $OX+{\angle AOX\over OX}$, where $\angle AOX$ is measured in radians in the range $[0,2\pi)$. Prove that we can find a point $X$, not on $OA$, such that its color appears on the circumference of the circle $C(X)$.

For a given positive integer $n >2$, let $C_{1},C_{2},C_{3}$ be the boundaries of three convex $n-$ gons in the plane , such that $C_{1}\cap C_{2}, C_{2}\cap C_{3},C_{1}\cap C_{3}$ are finite. Find the maximum number of points of the sets $C_{1}\cap C_{2}\cap C_{3}$.

Each side face of a dodecahedron lies in a uniquely determined plane. Those planes cut the space in a finite number of disjoint [i]regions[/i]. Find the number of such regions.

Given an unit square and some circles inside. Radius of each circle is less than $0.001$, and there is no couple of points belonging to the different circles with the distance between them $0.001$ exactly. Prove that the area, covered by the circles is not greater than $0.34$.

Let $ u_1, u_2, \ldots, u_m$ be $ m$ vectors in the plane, each of length $ \leq 1,$ with zero sum. Show that one can arrange $ u_1, u_2, \ldots, u_m$ as a sequence $ v_1, v_2, \ldots, v_m$ such that each partial sum $ v_1, v_1 \plus{} v_2, v_1 \plus{} v_2 \plus{} v_3, \ldots, v_1, v_2, \ldots, v_m$ has length less than or equal to $ \sqrt {5}.$

For each natural number $n\ge3$, let $m(n)$ be the maximum number of points inside or on the sides of a regular $n$-agon of side $1$ such that the distance between any two points is greater than $1$. Prove that $m(n)\ge n$ for $n>6$.

Prove that from any five points in the plane it is possible to choose three points that are not vertices of an acute triangle.

A connected checkered figure is drawn on a checkered paper. It is known that the figure can be cut both into $2\times 2$ squares and into (possibly rotated) [url=https://upload.wikimedia.org/wikipedia/commons/thumb/3/30/Tetromino-skew2.svg/1200px-Tetromino-skew2.svg.png]skew-tetrominoes[/url]. Prove that there is a hole in the figure. [i]Proposed by Y. Markelov and A. Sairanov[/i]

All internal angles of a convex octagon $ABCDEFGH$ are equal to each other and the edges are alternatively equal: $$AB = CD = EF = GH,BC = DE = FG = HA$$ (we call such an octagon semiregular). The diagonals $AD$, $BE$, $CF$, $DG$, $EH$, $FA$, $GB$ and $HC$ divide the inside of the octagon into certain parts. Consider the part containing the centre of the octagon. If that part is an octagon, then this central octagon is semiregular (this is obvious). In this case we construct similar diagonals in the central octagon and so on. If, after several steps, the central figure is not an octagon, then the process stops. Prove that if the process never stops, then the initial octagon was regular. (A. Tolpygo, Kiev)

There are $2017$ points on the plane, no three of them are collinear. Some pairs of the points are connected by $n$ segments. Find the smallest value of $n$ so that there always exists two disjoint segments in any case.

Suppose that $S$ is a finite set of points in the plane such that the area of triangle $\triangle ABC$ is at most $1$ whenever $A,B,$ and $C$ are in $S.$ Show that there exists a triangle of area $4$ that (together with its interior) covers the set $S.$

The plane has been partitioned into $N$ regions by three bunches of parallel lines. What is the least number of lines needed in order that $N > 1981$?

What is the maximal number of regions a circle can be divided in by segments joining $n$ points on the boundary of the circle ? [i]Posted already on the board I think...[/i]