Olympiad problems

2002 Junior Balkan Team Selection Tests - Moldova, 2

$64$ distinct points are positioned in the plane so that they determine exactly $2003$ different lines. Prove that among the $64$ points there are at least $4$ collinear points.

1999 Tournament Of Towns, 3

Tags: line , combinatorial geometry , combinatorics

There are $n$ straight lines in the plane such that each intersects exactly $1999$ of the others . Find all posssible values of $n$. (R Zhenodarov)

2011 Sharygin Geometry Olympiad, 18

Tags: combinatorial geometry , line , dissecting plane , geometry

On the plane, given are $n$ lines in general position, i.e. any two of them aren’t parallel and any three of them don’t concur. These lines divide the plane into several parts. What is a) the minimal, b) the maximal number of these parts that can be angles?

2009 Germany Team Selection Test, 2

Tags: geometry , point set , combinatorial geometry , line

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]

2013 Greece Team Selection Test, 4

Tags: combinatorics , counting , circles , line

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.

1967 Swedish Mathematical Competition, 1

Tags: combinatorial geometry , combinatorics , rectangle , line

$p$ parallel lines are drawn in the plane and $q$ lines perpendicular to them are also drawn. How many rectangles are bounded by the lines?

1996 Chile National Olympiad, 3

Tags: combinatorial geometry , combinatorics , line

Let $n> 2$ be a natural. Given $2n$ points in the plane, no $3$ are collinear. What is the maximum number of lines that can be drawn between them, without forming a triangle? [hide=original wording]Sea n > 2 un natural. Dados 2n puntos en el plano, tres a tres no colineales, Cual es el numero maximo de trazos que pueden dibujarse entre ellos, sin formar un triangulo?[/hide]

1979 Poland - Second Round, 4

Tags: geometry , symmetry , 3d geometry , line

Let $ S_k $ be the symmetry of the plane with respect to the line $ k $. Prove that equality holds for every lines $ a, b, c $ contained in one plane $$ S_aS_bS_cS_aS_bS_cS_bS_cS_aS_bS_cS_a = S_bS_cS_aS_bS_cS_aS_aS_bS_cS_aS_bS_c$$

1976 Vietnam National Olympiad, 5

Tags: geometry , 3d geometry , 3-dimensional geometry , line , distance

$L, L'$ are two skew lines in space and $p$ is a plane not containing either line. $M$ is a variable line parallel to $p$ which meets $L$ at $X$ and $L'$ at $Y$. Find the position of $M$ which minimises the distance $XY$. $L''$ is another fixed line. Find the line $M$ which is also perpendicular to $L''$ .

1948 Moscow Mathematical Olympiad, 147

Tags: combinatorial geometry , geometry , line

Consider a circle and a point $A$ outside it. We start moving from $A$ along a closed broken line consisting of segments of tangents to the circle (the segment itself should not necessarily be tangent to the circle) and terminate back at $A$. (On the links of the broken line are solid.) We label parts of the segments with a plus sign if we approach the circle and with a minus sign otherwise. Prove that the sum of the lengths of the segments of our path, with the signs given, is zero. [img]https://cdn.artofproblemsolving.com/attachments/3/0/8d682813cf7dfc88af9314498b9afcecdf77d2.png[/img]

2011 IFYM, Sozopol, 4

Tags: point , geometry , plane , line

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).

2001 239 Open Mathematical Olympiad, 6

Tags: combinatorics , line

On the plane 100 lines are drawn, among which there are no parallel lines. From any five of these lines, some three pass through one point. Prove that there are two points such that each line contains at least of of them.

1981 Austrian-Polish Competition, 8

Tags: combinatorial geometry , combinatorics , plane , line , parallel

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 $N > 1981$?

1987 Tournament Of Towns, (153) 4

Tags: line , line construction , geometry , arc , broken line , bisects area

We are given a figure bounded by arc $AC$ of a circle, and a broken line $ABC$, with the arc and broken line being on opposite sides of the chord $AC$. Construct a line passing through the mid-point of arc $AC$ and dividing the area of the figure into two regions of equal area.

1947 Moscow Mathematical Olympiad, 126

Tags: combinatorial geometry , combinatorics , convex , pentagon , line

Given a convex pentagon $ABCDE$, prove that if an arbitrary point $M$ inside the pentagon is connected by lines with all the pentagon’s vertices, then either one or three or five of these lines cross the sides of the pentagon opposite the vertices they pass. Note: In reality, we need to exclude the points of the diagonals, because that in this case the drawn lines can pass not through the internal points of the sides, but through the vertices. But if the drawn diagonals are not considered or counted twice (because they are drawn from two vertices), then the statement remains true.

1991 All Soviet Union Mathematical Olympiad, 537

Tags: combinatorial geometry , line , integer sides

Four lines in the plane intersect in six points. Each line is thus divided into two segments and two rays. Is it possible for the eight segments to have lengths $1, 2, 3, ... , 8$? Can the lengths of the eight segments be eight distinct integers?

2011 Oral Moscow Geometry Olympiad, 2

Tags: 3d geometry , geometry , line , plane

Line $\ell $ intersects the plane $a$. It is known that in this plane there are $2011$ straight lines equidistant from $\ell$ and not intersecting $\ell$. Is it true that $\ell$ is perpendicular to $a$?

1985 All Soviet Union Mathematical Olympiad, 406

Tags: line , combinatorial geometry , maximum

$n$ straight lines are drawn in a plane. They divide the plane onto several parts. Some of the parts are painted. Not a pair of painted parts has non-zero length common bound. Prove that the number of painted parts is not more than $\frac{n^2 + n}{3}$.

2003 Oral Moscow Geometry Olympiad, 6

Tags: combinatorics , line , geometry

A circle is located on the plane. What is the smallest number of lines you need to draw so that, symmetrically reflecting a given circle relative to these lines (in any order a finite number of times), it could cover any given point of the plane?

1998 Singapore MO Open, 2

Tags: line , lattice , collinear , function , inequalities , combinatorial geometry , combinatorics

Let $N$ be the set of natural numbers, and let $f: N \to N$ be a function satisfying $f(x) + f(x + 2) < 2 f(x + 1)$ for any $x \in N$. Prove that there exists a straight line in the $xy$-plane which contains infinitely many points with coordinates $(n,f(n))$.

1949-56 Chisinau City MO, 58

Tags: combinatorial geometry , combinatorics , line , point , geometry

On the plane $n$ points are chosen so that exactly $m$ of them lie on one straight line and no three points not included in these $m$ points lie on one straight line. What is the number of all lines, each of which contains at least two of these points?

2011 Tournament of Towns, 5

Tags: sum , combinatorial geometry , combinatorics , line , angle

In the plane are $10$ lines in general position, which means that no $2$ are parallel and no $3$ are concurrent. Where $2$ lines intersect, we measure the smaller of the two angles formed between them. What is the maximum value of the sum of the measures of these $45$ angles?

EGMO 2017, 3

Tags: line , plane , combinatorics , combinatorial geometry

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?

1971 Poland - Second Round, 3

Tags: geometry , combinatorics , combinatorial geometry , 3d geometry , line

There are 6 lines in space, of which no 3 are parallel, no 3 pass through the same point, and no 3 are contained in the same plane. Prove that among these 6 lines there are 3 mutually oblique lines.

2005 Sharygin Geometry Olympiad, 18

Tags: symmetry , line , circumcenter , concurrency , geometry

On the plane are three straight lines $\ell_1, \ell_2,\ell_3$, forming a triangle, and the point $O$ is marked, the center of the circumscribed circle of this triangle. For an arbitrary point X of the plane, we denote by $X_i$ the point symmetric to the point X with respect to the line $\ell_i, i = 1,2,3$. a) Prove that for an arbitrary point $M$ the straight lines connecting the midpoints of the segments $O_1O_2$ and $M_1M_2, O_2O_3$ and $M_2M_3, O_3O_1$ and $M_3M_1$ intersect at one point, b) where can this intersection point lie?