Given triangle $\vartriangle A_1A_2A_3$ and a straight line $\ell$ outside it. The angles between the lines $A_1A_2$ and $A_2A_3, A_1A_2$ and $A_2A_3, A_2A_3$ and $A_3A_1$ are equal to $a_3, a_1$ and $a_2$, respectively. The straight lines are drawn through points $A_1, A_2, A_3$ forming with $\ell$ angles of $\pi -a_1, \pi -a_2, \pi -a_3$, respectively. All angles are counted in the same direction from $\ell$ . Prove that these new lines meet at one point.

$N$ lines are drawn on the plane that divide it into a certain number for finite and endless parts. For which number of straight lines $n$ can there be more finite than infinite among the resulting level parts?

Let there be a finite number of straight lines in the plane, none of which are three in one point to cut. Show that the intersections of these straight lines can be colored with $3$ colors so that that no two points of the same color are adjacent on any of the straight lines. (Two points of intersection are called [i]adjacent [/i] if they both lie on one of the finitely many straight lines and there is no other such intersection on their connecting line.)

Six straight lines are given in space. Among any three of them, two are perpendicular. Show that the given lines can be labeled $\ell_1,...,\ell_6$ in such a way that $\ell_1, \ell_2, \ell_3$ are pairwise perpendicular, and so are $\ell_4, \ell_5, \ell_6$.

Which convex domains (figures) on a plane can contain an entire straight line? It is assumed that the figure is flat and does not degenerate into a straight line and is closed, that is, it contains all its boundary points.

Let $a$ and $b$ be given positive real numbers, with $a<b.$ If two points are selected at random from a straight line segment of length $b,$ what is the probability that the distance between them is at least $a?$

Given the intersecting lines $ r$ and $s$, consider the lines $u$ and $v$ as such what: a) $u$ is symmetric to $r$ with respect to $s$, b) $v$ is symmetric to $s$ with respect to $r$ . Determine the angle that the given lines must form such that $u$ and $v$ to be coplanar.

Given are $n$ different concentric circles on the plane.Inside the disk with the smallest radius (strictly inside it),we consider two distinct points $A,B$.We consider $k$ distinct lines passing through $A$ and $m$ distinct lines passing through $B$.There is no line passing through both $A$ and $B$ and all the lines passing through $k$ intersect with all the lines passing through $B$.The intersections do not lie on some of the circles.Determine the maximum and the minimum number of regions formed by the lines and the circles and are inside the circles.

Find the greatest $n$ for which there exist $n$ lines in space, passing through a single point, such that any two of them form the same angle.

$2019$ point grasshoppers sit on a line. At each move one of the grasshoppers jumps over another one and lands at the point the same distance away from it. Jumping only to the right, the grasshoppers are able to position themselves so that some two of them are exactly $1$ mm apart. Prove that the grasshoppers can achieve the same, jumping only to the left and starting from the initial position. (Sergey Dorichenko)

Suppose $n$ straight lines are in the plane so that there exist seven points such that any of these line passes through at least three of these points. Find the largest possible value of $n$.

Prove that for every set of $2n$ lines in the plane, such that there are no two parallel lines, there are two lines that divide the plane into four quadrants such that in each quadrant the number of unbounded regions is equal to $n$. [asy] unitsize(1cm); pair[] A, B; pair P, Q, R, S; A[1] = (0,5.2); B[1] = (6.1,0); A[2] = (1.5,5.5); B[2] = (3.5,0); A[3] = (6.8,5.5); B[3] = (1,0); A[4] = (7,4.5); B[4] = (0,4); P = extension(A[2],B[2],A[4],B[4]); Q = extension(A[3],B[3],A[4],B[4]); R = extension(A[1],B[1],A[2],B[2]); S = extension(A[1],B[1],A[3],B[3]); fill(P--Q--S--R--cycle, palered); fill(A[4]--(7,0)--B[1]--S--Q--cycle, paleblue); draw(A[1]--B[1]); draw(A[2]--B[2]); draw(A[3]--B[3]); draw(A[4]--B[4]); label("Bounded region", (3.5,3.7), fontsize(8)); label("Unbounded region", (5.4,2.5), fontsize(8)); [/asy]

Consider a circle centered at $O$ with radius $r$ and a line $\ell$ not passing through $O$. A grasshopper is jumping to and fro between the points of the circle and the line, the length of each jump being $r$. Prove that there are at most $8$ points for the grasshopper to reach.

There are $19$ lines in the plane dividing the plane into exactly $97$ pieces. (a) Prove that among these pieces there is at least one triangle. (b) Show that it is indeed possible to place $19$ lines in the above way.

Let $n$ be a fixed positive integer and fix a point $O$ in the plane. There are $n$ lines drawn passing through the point $O$. Determine the largest $k$ (depending on $n$) such that we can always color $k$ of the $n$ lines red in such a way that no two red lines are perpendicular to each other. [i]Proposed by Nikola Velov[/i]

Let $ k$ and $ n$ be integers with $ 0\le k\le n \minus{} 2$. Consider a set $ L$ of $ n$ lines in the plane such that no two of them are parallel and no three have a common point. Denote by $ I$ the set of intersections of lines in $ L$. Let $ O$ be a point in the plane not lying on any line of $ L$. A point $ X\in I$ is colored red if the open line segment $ OX$ intersects at most $ k$ lines in $ L$. Prove that $ I$ contains at least $ \dfrac{1}{2}(k \plus{} 1)(k \plus{} 2)$ red points. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

Let $k$ be a circle with center $M.$ There is another circle $k_1$ whose center $M_1$ lies on $k,$ and we denote the line through $M$ and $M_1$ by $g.$ Let $T$ be a point on $k_1$ and inside $k.$ The tangent $t$ to $k_1$ at $T$ intersects $k$ in two points $A$ and $B.$ Denote the tangents (diifferent from $t$) to $k_1$ passing through $A$ and $B$ by $a$ and $b$, respectively. Prove that the lines $a,b,$ and $g$ are either concurrent or parallel.

We drew several straight lines on the plane and marked all of them intersection points. How many lines could be drawn? if one point is marked on one of the drawn lines, on the other - three, and on the third - five? Find all possible options and prove that there are no others.

A set of lines in the plan is called [i]nice [/i]i f every line in the set intersects an odd number of other lines in the set. Determine the smallest integer $k \ge 0$ having the following property: for each $2018$ distinct lines $\ell_1, \ell_2, ..., \ell_{2018}$ in the plane, there exist lines $\ell_{2018+1},\ell_{2018+2}, . . . , \ell_{2018+k}$ such that the lines $\ell_1, \ell_2, ..., \ell_{2018+k}$ are distinct and form a [i]nice [/i] set.

Given the point $O$ in the space and $1979$ straight lines $l_1, l_2, ... , l_{1979}$ containing it. Not a pair of lines is orthogonal. Given a point $A_1$ on $l_1$ that doesn't coincide with $O$. Prove that it is possible to choose the points $A_i$ on $l_i$ ($i = 2, 3, ... , 1979$) in so that $1979$ pairs will be orthogonal: $A_1A_3$ and $l_2$, $A_2A_4$ and $l_3$,$ ...$ , $A_{i-1}A_{i+1}$ and $l_i$,$ ...$ , $A_{1977}A_{1979}$ and $l_{1978}$, $A_{1978}A_1$ and $l_{1979}$, $A_{1979}A_2$ and $l_1$

Determine the minimum number of lines that can be drawn on the plane so that they intersect in exactly $200$ distinct points. (Note that for $3$ distinct points, the minimum number of lines is $3$ and for $4$ distinct points, the minimum is $4$)

Three straight lines $\ell_1$, $\ell_2$ and $\ell_3$, forming a triangle, divide the plane into $7$ parts. Each of the points $M_1$, $M_2$ and $M_3$ lies in one of the angles, vertical to some angle of the triangle. The distance from $M_1$ to straight lines $\ell_1$, $\ell_2$ and $\ell_3$ are equal to $7,3$ and $1$ respectively The distance from $M_2$ to the same lines are $4$, $1$ and $3$ respectively. For $M_3$ these distances are $3$, $5$ and $2$. What is the radius of the circle inscribed in the triangle? [hide=second sentence in Russian]Каждая из точек М_1, М_2 и М_з лежит в одном из углов, вертикальном по отношению к какому-то углу треугольника.[/hide]

Let a real number $k$ and points $S,A,SA=1$ in plane be given. Denote $A'$ the image of $A$ under rotation by an oriented angle $\varphi$ with respect to center $S$. Similarly, let $A''$ be the image of $A'$ under homothety with the factor $\frac{1}{\cos\varphi-k\sin\varphi}$ with respect to center $S.$ Denote the locus \[\ell=\bigl\{A''\mid\varphi\in(-\pi,\pi],\cos\varphi-k\sin\varphi\neq0\bigr\}.\] Show that $\ell$ is a line containing $A.$

In the plane, $n$ lines are drawn in general position (that is, there are neither two of them parallel nor three of them passing through the same point). Prove that it is possible to put a positive integer in each region (finite or infinite) determined by these lines so that for each line the sum of the numbers in the regions of a sdemiplane is equal to the sum of the numbers in the regions of the other semiplane. Note: A region is a set of points such that the straight line connecting any two of them it does not intersect any of the lines. For example, a line divides the plane into $2$ infinite regions and three lines into general position divide the plane into $7$ regions, some finite(s) and others infinite.

Given a triangle $ABC$ with an interior point $P$ and points $Q_1 , Q_2$ not lying on any of the segments $AB , AC ,BC,$ $AP ,BP ,CP,$ show that there does not exist a polygonal line $K$ joining $Q_1$ and $Q_2$ such that i) $K$ crosses each segment exactly once, ii) $K$ does not intersect itself iii) $K$ does not pass through $A, B , C$ or $P.$