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.

The pyramid $VABCD$ has base the rectangle ABCD, and the side edges are congruent. Prove that the plane $(VCD)$ forms congruent angles with the planes $(VAC)$ and $(BAC)$ if and only if $\angle VAC = \angle BAC $.

Is it possible that two cross-sections of a tetrahedron by two different cutting planes are two squares, one with a side of length no greater than $1$ and another with a side of length at least $100$? Mikhail Evdokimov

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.

All edges of the triangular pyramid $ABCD$ are equal in length. Let $M$ be the midpoint of $DB$, $N$ is the point on $\overline{AB}$, such that $2NA=NB$ and $N\not\in AB$ and $P$ is a point on the altitude through point $D$ in $\triangle BCD$. Find $\angle MPD$ if the intersection of the pyramid with the plane $(NMP)$ is a trapezoid.

Let $A_1A_2A_3A_4$ be a tetrahedron. For any permutation $(i, j,k,h)$ of $1,2,3,4$ denote: - $P_i$ – the orthogonal projection of $A_i$ on $A_jA_kA_h$; - $B_{ij}$ – the midpoint of the edge $A_iAj$, - $C_{ij}$ – the midpoint of segment $P_iP_j$ - $\beta_{ij}$– the plane $B_{ij}P_hP_k$ - $\delta_{ij}$ – the plane $B_{ij}P_iP_j$ - $\alpha_{ij}$ – the plane through $C_{ij}$ orthogonal to $A_kA_h$ - $\gamma_{ij}$ – the plane through $C_{ij}$ orthogonal to $A_iA_j$. Prove that if the points $P_i$ are not in a plane, then the following sets of planes are concurrent: (a) $\alpha_{ij}$, (b) $\beta_{ij}$, (c) $\gamma_{ij}$, (d) $\delta_{ij}$.

Each of two regular polyhedrons $P$ and $Q$ was divided by the plane into two parts. One part of $P$ was attached to one part of $Q$ along the dividing plane and formed a regular polyhedron not equal to $P$ and $Q$. How many faces can it have?

A triangle $ABC$ and spheres are given in space $S_1$ and $S_2$, each of which passes through points $A, B$ and $C$. For points $M$ spheres $S_1$ not lying in the plane of triangle $ABC$ are drawn lines $MA, MB$ and $MC$, intersecting the sphere $S_2$ for the second time at points $A_1,B_1$ and $C_1$, respectively. Prove that the planes passing through points $A_1, B_1$ and $C_1$, touch a fixed sphere or pass through a fixed point.

There are $2017$ lines in the plane such that no three of them go through the same point. Turbo the snail sits on a point on exactly one of the lines and starts sliding along the lines in the following fashion: she moves on a given line until she reaches an intersection of two lines. At the intersection, she follows her journey on the other line turning left or right, alternating her choice at each intersection point she reaches. She can only change direction at an intersection point. Can there exist a line segment through which she passes in both directions during her journey?

A grasshopper hops from an integer point to another integer point in the plane, where every even jump has a length $\sqrt{98}$ and every odd one - $\sqrt{149}$. What’s the least number of jumps the grasshopper has to make in order to return to its starting point after odd number of jumps?

Consider the cube $ABCDA'B'C'D'$. The bisectors of the angles $\angle A' C'A$ and $\angle A' AC'$ intersect $AA'$ and $A'C$ in the points $P$, respectively $S$. The point $M$ is the foot of the perpendicular from $A'$ on $CP$ , and $N$ is the foot of the perpendicular from $A'$ to $AS$. Point $O$ is the center of the face $ABB'A'$ a) Prove that the planes $(MNO)$ and $(AC'B)$ are parallel. b) Calculate the distance between these planes, knowing that $AB = 1$.

Determine the set $S$ with minimal number of points defining $7$ distinct lines

Suppose that each set $X$ of points in the plane has an associated set $\overline{X}$ of points called its cover. Suppose further that (1) $\overline{X\cup Y} \supset \overline{\overline{X}} \cup \overline{Y} \cup Y$ for all sets $X,Y$ . Show that i) $\overline{X} \supset X$, ii) $\overline{\overline{X}}=\overline{X}$ and iii) $X\supset Y \Rightarrow \overline{X} \supset \overline{Y}.$ Prove also that these three statements imply (1).

A tetrahedron $ABCD$ is given. The incircles of triangles $ ABC$ and $ABD$ with centers $O_1, O_2$, touch $AB$ at points $T_1, T_2$. The plane $\pi_{AB}$ passing through the midpoint of $T_1T_2$ is perpendicular to $O_1O_2$. The planes $\pi_{AC},\pi_{BC}, \pi_{AD}, \pi_{BD}, \pi_{CD}$ are defined similarly. Prove that these six planes have a common point.

A plane passing through the center of a cube intersects the cube in a cyclic hexagon. Show that this hexagon is regular.

In $3$ dimensional space we are given a parallelogram $ABCD$ and plane $M$. The distances from vertices $A, B$ and $C$ to plane $M$ are $a, b$ and $c$ respectively. Find the distance $d$ from vertex $D$ to the plane $M$ .

Let $OP$ be a diagonal of a unit cube. Find the minimum and the maximum value of the area of the intersection of the cube with a plane through $OP$.

Let $M$ be a finite subset of the plane such that for any two different points $A,B\in M$ there is a point $C\in M$ such that $ABC$ is equilateral. What is the maximal number of points in $M?$

It is given $2018$ points in plane. Prove that it is possible to cover them with circles such that: $i)$ sum of lengths of all diameters of all circles is not greater than $2018$ $ii)$ distance between any two circles is greater than $1$

In the plane $n$ segments with lengths $a_1, a_2, \dots , a_n$ are drawn. Every ray beginning at the point $O$ meets at least one of the segments. Let $h_i$ be the distance from $O$ to the $i$-th segment (not the line!) Prove the inequality \[\frac{a_1}{h_1}+\frac{a_2}{h_2} + \ldots + \frac{a_i}{h_i} \geqslant 2 \pi.\]

For non-coplanar points are given in space. A plane $\pi$ is called [i]equalizing [/i] if all four points have the same distance from $\pi$. Find the number of equilizing planes.

Anne has thought of a finite set $A \subseteq \mathbb{R}^2 $ . Bob does not know how many elements $A$ has, but his goal is to completely determine $A$. To achieve this, Bob can chooseany point $b \in \mathbb{R}^2 $ and ask Anne how far it is from$ A$ . Anne replies with the distance, defined as $min \{d(a, b) | a \in A\}$. (Here $d(a, b)$ denotes the distance between points $a, b \in \mathbb{R}$ .) Bob can ask as many questions of this type as he wants, until he can determine A with certainty. a) Can Bob achieve his goal with finitely many questions? b) What if Anne tells Bob in advance that all points of A have both coordinates in the interval$\ [0, 1]\ $? Note: $\mathbb{R}^2$ is the set of points in the plane.

There are $n$ points in a plane. Prove that there exist a point $O$ (not necessarily from the given $n$) such that on each side of an arbitrary line, through $O$, lie at least $\frac{n}{3}$ points (including the points on the line).

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 $N > 1981$?

A plane intersects a sphere in a circle $C$. The points $A$ and $B$ lie on the sphere on opposite sides of the plane. The line joining $A$ to the center of the sphere is normal to the plane. Another plane $p$ intersects the segment $AB$ and meets $C$ at $P$ and $Q$. Show that $BP\cdot BQ$ is independent of the choice of $p$.