Suppose $28$ 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?

a) Given $5$ points on a plane, no three of which lie on one line. Prove that four of these points can be taken as vertices of a convex quadrilateral. b) Inside a square, consider a convex quadrilateral and inside the quadrilateral, take a point $A$. It so happens that no three of the $9$ points — the vertices of the square, of the quadrilateral and $A$ — lie on one line. Prove that $5$ of these points are vertices of a convex pentagon.

Select $ n$ points on a circle independently with uniform distribution. Let $ P_n$ be the probability that the center of the circle is in the interior of the convex hull of these $ n$ points. Calculate the probabilities $ P_3$ and $ P_4$. [A. Renyi]

Seven points are selected randomly from $ S^1\subset\mathbb C$. What is the probability that origin is not contained in convex hull of these points?

Given $1980$ vectors in the plane, and there are some non-collinear among them. The sum of every $1979$ vectors is collinear to the vector not included in that sum. Prove that the sum of all vectors equals to the zero vector.

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 $B$.

The centre of a circle is the origin of $N$ vectors whose ends divide the circle in $N$ equal arcs . Some of the vectors are blue and some are red . We calculate the sum of the angles formed between each pair consisting of a red vector and a blue vector (the angle being measured anticlockwise from red to blue) and divide this sum by the total number of such angles . Prove that the "mean angle" thus obtained is $180^o$. (V. P roizvolov)

Can one draw , on the surface of a Rubik's cube , a closed path which crosses each little square exactly once and does not pass through any vertex of a square? (S . Fomin, Leningrad)

On a plane several straight lines are drawn in such a way that each of them intersects exactly $15$ other lines. How many lines are drawn on the plane? Find all possibilities and justify your answer. Poland

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 $4$.

Julián chooses $2007$ points of the plane between which there are no $3$ aligned, and draw with red all the segments that join two of those points. Next, Roberto draws several lines. Its objective is for each red segment to be cut inside by (at least) one of the lines. Determine the minor $\ell$ lines such that, no matter how Julián chooses the $2007$ points, with the properly chosen $\ell$ lines, Roberto will achieve his objective with certainty.

Each side of a triangle $ABC$ has been divided into $n+1$ equal parts. Find the number of triangles with the vertices at the division points having no side parallel to or lying at a side of $\triangle ABC$.

A straight line is drawn across an $8\times 8$ chessboard. It is said to [i]pierce [/i]a square if it passes through an interior point of the square. At most how many of the $64$ squares can this line [i]pierce[/i]?

Prove that if every line connecting any two points of some finite set of points of the plane contains at least one more point of this set, then all points of the set lie on one straight line.

There are 100 blue lines drawn on the plane, among which there are no parallel lines and no three of which pass through one point. The intersection points of the blue lines are marked in red. Could it happen that the distance between any two red dots lying on the same blue line is equal to an integer? [i]From the folklore[/i]

Is it possible to place $100$ balls in space so that no two of them have a common interior point and each of them touches at least one third of the others?

A rectangular grid whose side lengths are integers greater than $1$ is given. Smaller rectangles with area equal to an odd integer and length of each side equal to an integer greater than $1$ are cut out one by one. Finally one single unit is left. Find the least possible area of the initial grid before the cuttings. Ps. Collected [url=https://artofproblemsolving.com/community/c949611_2019_nigerian_senior_mo_round_3]here[/url]

Three colors are available to paint the plane. If each point in the plane is assigned one of these three colors, prove that there is a segment of length $1$ whose endpoints have the same color.

In the plane are given $1997$ points. Show that among the pairwise distances between these points, there are at least $32$ different values.

A numismatist Fred has some coins. A diameter of any coin is no more than $10$ cm. All the coins are contained in a one-layer box of dimensions $30$ cm by $70$ cm. He is presented with a new coin. Its diameter is $25$ cm. Prove that it is possible to put all the coins in a one-layer box of dimensions $55$ cm by $55$ cm. (Fedja Nazarov, St Petersburg)

A convex polygon with $2n$ sides is called [i]rhombic [/i] if its sides are equal and all pairs of opposite sides are parallel. A rhombic polygon can be partitioned into rhombic quadrilaterals. For what value of$ n$, a $2n$-sided rhombic polygon splits into $666$ rhombic quadrilaterals?

Given a $101\times 200$ sheet of graph paper, we start moving from a corner square in the direction of the square’s diagonal (not the sheet’s diagonal) to the border of the sheet, then change direction obeying the laws of light’s reflection. Will we ever reach a corner square? [img]https://cdn.artofproblemsolving.com/attachments/b/8/4ec2f4583f406feda004c7fb4f11a424c9b9ae.png[/img]

How many diagonals can you draw in a convex $2009$-gon if in the finished drawing, every drawn diagonal inside the $2009$-gon may cut at most another drawn diagonal?

Alice and Bob play a game, in which they take turns drawing segments of length $1$ in the Euclidean plane. Alice begins, drawing the first segment, and from then on, each segment must start at the endpoint of the previous segment. It is not permitted to draw the segment lying over the preceding one. If the new segment shares at least one point - except for its starting point - with one of the previously drawn segments, one has lost. a) Show that both Alice and Bob could force the game to end, if they don’t care who wins. b) Is there a winning strategy for one of them?

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.