Suppose \(40\) objects are placed along a circle at equal distances. In how many ways can \(3\) objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?

For what values of $n$ is it possible to mark $n$ points on the plane so that each point has at least three other points at distance $1$?

One day, Professor Umzar Azum decided to fry dumplings for dinner. He took out a frying pan, opened a pack of dumplings, but suddenly thought about the question: how many dumplings could he fit in his frying pan? Measuring the sizes of the frying pan and dumplings, the professor came to the conclusion that the dumplings have the shape of a semicircle, the diameter of which is four times smaller than the diameter of the frying pan. Show how on the frying pan it is possible to place (without overlap): a) $20$ pieces of dumplings; b) $24$ pieces of dumplings; . (The problem boils down to placing, without overlapping, the appropriate number of identical semicircles inside a circle with a diameter four times larger.) [i]Note: We (the authors of the problem) do not know the answer to the question whether it is possible to place 25 semicircles in a circle with a diameter four times smaller, and even more so we do not know what the largest number of such semicircles is. We will welcome any progress in solving the problem and evaluate it accordingly. [/i]

A convex quadrilateral of area $1$ is given. Prove that there exist four points in the interior or on the sides of the quadrilateral such that each triangle with the vertices in three of these four points has an area greater than or equal to $1/4$.

Each point in a $12 \times 12$ array is colored red, white or blue. Show that it is always possible to find $4$ points of the same color forming a rectangle with sides parallel to the sides of the array.

Supppose five points in a plane are situated so that no two of the straight lines joining them are parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all the lines joining the other four points. Determine the maxium number of intersections that these perpendiculars can have.

There are $n$ distinct lines in three-dimensional space such that no two lines are parallel and no three lines meet at one point. What is the maximal possible number of planes determined by these $n$ lines? We say that a plane is determined if it contains at least two of the lines.

The area of a convex polygon in the plane is equally shared by the four standard quadrants, and all non-zero lattice points lie outside the polygon. Show that the area of the polygon is less than $4$.

A convex $1994$-gon $M$ is given in the plane. A closed polygonal line consists of $997$ of its diagonals. Every vertex is adjacent to exactly one diagonal. Each diagonal divides $M$ into two sides, and the smaller of the numbers of edges on the two sides of $M$ is defined to be the length of the diagonal. Is it posible to have (a) $991$ diagonals of length $3$ and $6$ of length $2$? (b) $985$ diagonals of length $6, 4$ of length $8$, and $8$ of length $3$?

There are distinct points $O, A, B, K_1, . . . , K_n, L_1, . . . , L_n$ on a plane such that no three points are collinear. The open line segments $K_1L_1, . . . , K_nL_n$ are coloured red, other points on the plane are left uncoloured. An allowed path from point $O$ to point $X$ is a polygonal chain with first and last vertices at points $O$ and $X$, containing no red points. For example, for $n = 1$, and $K_1 = (-1, 0)$, $L_1 = (1, 0)$, $O = (0,-1)$, and $X = (0,1)$, $OK_1X$ and $OL_1X$ are examples of allowed paths from $O$ to $X$, there are no shorter allowed paths. Find the least positive integer n such that it is possible that the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $A$ is closer to $B$ than to $A$, and the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $B$ is closer to $A$ than to $B$.

What is the smallest number of disks radius $\frac12$ that can cover a disk radius $1$?

Find the smallest real $\alpha$, such that for any convex polygon $P$ with area $1$, there exist a point $M$ in the plane, such that the area of convex hull of $P\cup Q$ is at most $\alpha$, where $Q$ denotes the image of $P$ under central symmetry with respect to $M$.

Determine whether or nor there exist two disjoint infinite sets $ A$ and $ B$ of points in the plane satisfying the following conditions: a.) No three points in $ A \cup B$ are collinear, and the distance between any two points in $ A \cup B$ is at least 1. b.) There is a point of $ A$ in any triangle whose vertices are in $ B,$ and there is a point of $ B$ in any triangle whose vertices are in $ A.$

Let $P_i(x_i,y_i)\ (i=1,2,\cdots,2023)$ be $2023$ distinct points on a plane equipped with rectangular coordinate system. For $i\neq j$, define $d(P_i,P_j) = |x_i - x_j| + |y_i - y_j|$. Define $$\lambda = \frac{\max_{i\neq j}d(P_i,P_j)}{\min_{i\neq j}d(P_i,P_j)}$$. Prove that $\lambda \geq 44$ and provide an example in which the equality holds.

Given a figure, containing $16$ segments. You should prove that there is no curve, that intersect each segment exactly once. The curve may be not closed, may intersect itself, but it is not allowed to touch the segments or to pass through the vertices. [asy] draw((0,0)--(6,0)--(6,3)--(0,3)--(0,0)); draw((0,3/2)--(6,3/2)); draw((2,0)--(2,3/2)); draw((4,0)--(4,3/2)); draw((3,3/2)--(3,3)); [/asy]

There are several equal (possibly overlapping) square-shaped napkins on a rectangular table, with sides parallel to the sides of the table. Prove that it is possible to nail some of them to the table in such a way that every napkin is nailed exactly once.

There is a grid of equilateral triangles with a distance 1 between any two neighboring grid points. An equilateral triangle with side length $n$ lies on the grid so that all of its vertices are grid points, and all of its sides match the grid. Now, let us decompose this equilateral triangle into $n^2$ smaller triangles (not necessarily equilateral triangles) so that the vertices of all these smaller triangles are all grid points, and all these small triangles have equal areas. Prove that there are at least $n$ equilateral triangles among these smaller triangles.

A positive integer $n < 2017$ is given. Exactly $n$ vertices of a regular 2017-gon are colored red, and the remaining vertices are colored blue. Prove that the number of isosceles triangles whose vertices are monochromatic does not depend on the chosen coloring (but does depend on $n$.)

Consider a set of $n > 3$ points in the plane, no three of which are collinear, and a natural number $k < n$. Prove the following statements: (a) If $k \le \frac{n}{2}$, then each point can be connected with at least k other points by segments so that no three segments form a triangle. (b) If $k \ge \frac{n}{2}$, and each point is connected with at least k other points by segments, then some three segments form a triangle.

A total of $a \cdot b \cdot c$ cubical boxes are joined together in a $a \times b \times c$ rectangular stack, where $a, b, c \ge 2$. A bee is found inside one of the boxes. It can fly from one box to another through a hole in the wall, but not through edges or corners. Also, it cannot fly outside the stack. For which triples $(a, b, c)$ is it possible for the bee to fly through all of the boxes exactly once, and end up in the same box where it started?

Every cell of a $4\times 4$ square grid net, has $1\times 1$ size. Is it possible to represent this net as a union of the following sets: a) Eight broken lines of length five each? b) Five broken lines of length eight each?

Let $m,n>2$ be integers. A regular ${n}$-sided polygon region $\mathcal T$ on a plane contains a regular ${m}$-sided polygon region with a side length of ${}{}{}1$. Prove that any regular ${m}$-sided polygon region $\mathcal S$ on the plane with side length $\cos{\pi}/[m,n]$ can be translated inside $\mathcal T.$ In other words, there exists a vector $\vec\alpha,$ such that for each point in $\mathcal S,$ after translating the vector $\vec\alpha$ at that point, it fall into $\mathcal T.$ Note: The polygonal area includes both the interior and boundaries. [i]Created by Bin Wang[/i]

(a) $10$ points dividing a circle into $10$ equal arcs are connected in pairs by $5$ chords. Is it necessary that two of these chords are of equal length? (b) $20$ points dividing a circle into $20$ equal arcs are connected in pairs by $10$ chords. Prove that among these $10$ chords there are two chords of equal length. (VV Proizvolov, Moscow)

If $\overrightarrow{u_1},\overrightarrow{u_2}, ...,\overrightarrow{u_n}$ be vectors in the plane such that the sum of their lengths is at least $1$, then between them we find vectors whose sum is a vector of length at least $\sqrt2/8$. Prove it.

There are $2n+1$ segments on the line. Any segment intersects at with at least $n$ others. Prove that there is a segment that intersects all the others.