Let $m$ be a convex polygon in a plane, $l$ its perimeter and $S$ its area. Let $M\left( R\right) $ be the locus of all points in the space whose distance to $m$ is $\leq R,$ and $V\left(R\right) $ is the volume of the solid $M\left( R\right) .$ [i]a.)[/i] Prove that \[V (R) = \frac 43 \pi R^3 +\frac{\pi}{2} lR^2 +2SR.\] Hereby, we say that the distance of a point $C$ to a figure $m$ is $\leq R$ if there exists a point $D$ of the figure $m$ such that the distance $CD$ is $\leq R.$ (This point $D$ may lie on the boundary of the figure $m$ and inside the figure.) additional question: [i]b.)[/i] Find the area of the planar $R$-neighborhood of a convex or non-convex polygon $m.$ [i]c.)[/i] Find the volume of the $R$-neighborhood of a convex polyhedron, e. g. of a cube or of a tetrahedron. [b]Note by Darij:[/b] I guess that the ''$R$-neighborhood'' of a figure is defined as the locus of all points whose distance to the figure is $\leq R.$

Given $n>4$ points in the plane, no three collinear. Prove that there are at least $\frac{(n-3)(n-4)}{2}$ convex quadrilaterals with vertices amongst the $n$ points.

Which is the least positive integer $n$ for which it is possible to find a (non-degenerate) $n$-gon with sidelengths $1, 2,. . . , n$, and where all vertices have integer coordinates?

Six points are placed on the plane such that each three of them are the vertices of a triangle with sides of different lengths. Prove that the shortest side of one of these triangles is also the longest side of another of them.

$A_1B_1A_2$, $B_1A_2B_2$, $A_2B_2A_3$,...,$B_{13}A_{14}B_{14}$, $A_{14}B_{14}A_1$ 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?

Call the intersection of two segments [i]almost perfect[/i] if for each of the segments the distance between the midpoint of the segment and the intersection is at least $2022$ times smaller than the length of the segment. Prove that there exists a closed broken line of segments such that every segment intersects at least one other segment, and every intersection of segments is [i]almost perfect[/i].

The plan considers $51$ points of integer coordinates, so that the distances between any two points are natural numbers. Show that at least $49\%$ of the distances are even.

Consider an equilateral triangle with every side divided by $n$ points into $n+1$ equal parts. We put a marker on every of the $3n$ division points. We draw lines parallel to the sides of the triangle through the division points, and this way divide the triangle into $(n+1)^2$ smaller ones. Consider the following game: if there is a small triangle with exactly one vertex unoccupied, we put a marker on it and simultaneously take markers from the two its occupied vertices. We repeat this operation as long as it is possible. (a) If $n\equiv1\pmod3$, show that we cannot manage that only one marker remains. (b) If $n\equiv0$ or $n\equiv2\pmod3$, prove that we can finish the game leaving exactly one marker on the triangle.

We consider a number of points in a plane. Each of these points is connected to at least one of the other points by a line segment, in such a way that a figure arises that does not break up into different parts (that is, from any point along drawn line segments we can reach any other point).. We assign a point the ”order” $n$, when in this point $n$ line segments meet. We characterize the obtained figure by writing down the order of each of its points one after the other. For example, a hexagon is characterized by the combination $\{2,2,2,2,2,2\}$ and a star with six rays by $\{6,1,1,1,1,1,1\}$. (a) Sketch a figure' belonging to the combination $\{4,3,3,3,3\}$. (b) Give the combinations of all possible figures, of which the sum of the order numbers is equal to $6$. (c) Prove that every such combination contains an even number of odd numbers.

Is it possible to write natural numbers at all points of the plane with integer coordinates so that three points with integer coordinates lie on the same line if and only if the numbers written in them had a common divisor greater than one?

There are $1998$ rectangular pieces $2$ cm wide and $3$ cm long and with them squares are assembled (without overlapping or gaps). What is the greatest number of different squares that can be had at the same time?

Several (at least three) turtles are crawling along the plane, the velocities of which are constant in magnitude and direction (all are equal in magnitude, but pairwise different in direction). Prove that regardless of the initial location, after some time all the turtles will be at the vertices of some convex polygon.

Let $n$ be a natural number. Suppose $A$ and $B$ are two sets, each containing $n$ points in the plane, such that no three points of a set are collinear. Let $T(A)$ be the number of broken lines, each containing $n-1$ segments, and such that it doesn't intersect itself and its vertices are points of $A$. Define $T(B)$ similarly. If the points of $B$ are vertices of a convex $n$-gon (are in [i]convex position[/i]), but the points of $A$ are not, prove that $T(B)<T(A)$. [i]Proposed by Ali Khezeli[/i]

Given a regular $2013$-sided polygon, how many isosceles triangles are there whose vertices are vertices vertex of given polygon and haave an angle greater than $120^o$? Nguyen Tien Lam, High School for Natural Science,Hanoi National University.

Given a positive integer $n$, does there exist a planar polygon and a point in its plane such that every line through that point meets the boundary of the polygon at exactly $2n$ points?

Prove that for each $n \ge 4$ a parallelogram can be dissected in $n$ cyclic quadrilaterals.

Let $n$ be a positive integer. E. Chen and E. Chen play a game on the $n^2$ points of an $n \times n$ lattice grid. They alternately mark points on the grid such that no player marks a point that is on or inside a non-degenerate triangle formed by three marked points. Each point can be marked only once. The game ends when no player can make a move, and the last player to make a move wins. Determine the number of values of $n$ between $1$ and $2013$ (inclusive) for which the first player can guarantee a win, regardless of the moves that the second player makes. [i]Ray Li[/i]

Let $n>1$ be an integer. Suppose we are given $2n$ points in the plane such that no three of them are collinear. The points are to be labelled $A_1, A_2, \dots , A_{2n}$ in some order. We then consider the $2n$ angles $\angle A_1A_2A_3, \angle A_2A_3A_4, \dots , \angle A_{2n-2}A_{2n-1}A_{2n}, \angle A_{2n-1}A_{2n}A_1, \angle A_{2n}A_1A_2$. We measure each angle in the way that gives the smallest positive value (i.e. between $0^{\circ}$ and $180^{\circ}$). Prove that there exists an ordering of the given points such that the resulting $2n$ angles can be separated into two groups with the sum of one group of angles equal to the sum of the other group.

Show for any positive integer $n$ that there exists a circle in the plane such that there are exactly $n$ grid points within the circle. (A grid point is a point having integer coordinates.)

Given any $22$ points in the plane, no three collinear. Show that the points can be divided into $11$ pairs, so that the $11$ line segments defined by the pairs have at least five different intersections

An open chain was made from $54$ identical single cardboard squares, connecting them hingedly at the vertices. Any square (except for the extreme ones) is connected to its neighbors by two opposite vertices. Is it possible to completely cover a $3\times 3 \times3$ surface with this chain of squares?

One tiles a floor of $a \times b$ dm$^2$ with square tiles, $a,b \in N$. Tiles do not overlap, and sides of floor and tiles are parallel. Using tiles of $2\times 2$ dm$^2$ leaves the same amount of floor uncovered as using tiles of $4\times 4$ dm$^2$. Using $3\times 3$ dm$^2$ tiles leaves $29$ dm$^2$ floor uncovered. Determine $a$ and $b$.

Integers $1, 2,...,100$ are written on a circle, not necessarily in that order. Can it be that the absolute value of the dierence between any two adjacent integers is at least $30$ and at most $50$?

A convex$N-$gon with equal sides is located inside a circle. Each side is extended in both directions up to the intersection with the circle so that it contains two new segments outside the polygon. Prove that one can paint some of these new $2N$ segments in red and the rest in blue so that the sum of lengths of all the red segments would be the same as for the blue ones. [i]($8$ points)[/i]

The points inside a circle \( \Gamma \) are painted with \( n \geq 1 \) colors. A color is said to be dense in a circle \( \Omega \) if every circle contained within \( \Omega \) has points of that color in its interior. Prove that there exists at least one color that is dense in some circle contained within \( \Gamma \).