Olympiad problems

2006 Sharygin Geometry Olympiad, 13

Two straight lines $a$ and $b$ are given and also points $A$ and $B$. Point $X$ slides along the line $a$, and point $Y$ slides along the line $b$, so that $AX \parallel BY$. Find the locus of the intersection point of $AY$ with $XB$.

2013 Greece Team Selection Test, 4

Tags: counting , line , circles , combinatorics

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.

2009 Germany Team Selection Test, 2

Tags: point set , line , combinatorial geometry , geometry

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]

2011 Tournament of Towns, 5

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

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?

II Soros Olympiad 1995 - 96 (Russia), 10.7

Tags: line , geometry , inradius

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]

1998 German National Olympiad, 1

Tags: line , point , combinatorics , combinatorial geometry

Find all possible numbers of lines in a plane which intersect in exactly $37$ points.

2003 Oral Moscow Geometry Olympiad, 6

Tags: geometry , combinatorics , line

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?

1953 Moscow Mathematical Olympiad, 252

Tags: concurrency , angle , line , geometry

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.

1975 Czech and Slovak Olympiad III A, 4

Tags: analytic geometry , parameterization , linear algebra , absolute value , line , algebra

Determine all real values of parameter $p$ such that the equation \[|x-2|+|y-3|+y=p\] is an equation of a ray in the plane $xy.$

2008 Postal Coaching, 6

Tags: line , point , combinatorial geometry

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 $n$.

1996 Chile National Olympiad, 3

Tags: combinatorics , line , combinatorial geometry

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]

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.

1971 Poland - Second Round, 3

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

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.

2001 239 Open Mathematical Olympiad, 6

Tags: combinatorics , projective plane , block designs , line

On the plane 1000 lines are drawn, among which there are no parallel lines. From any seven of these lines, some three pass through one point. But no more than 500 lines pass through each point. Prove that there are three points such that each line contains at least of of them.

1962 Czech and Slovak Olympiad III A, 3

Tags: geometry , 3d geometry , line , locus , circles

Let skew lines $PM, QN$ be given such that $PM\perp PQ\perp QN$. Let a plane $\sigma\perp PQ$ containing the midpoint $O$ of segment $PQ$ be given and in it a circle $k$ with center $O$ and given radius $r$. Consider all segments $XY$ with endpoint $X, Y$ on lines $PM, QN$, respectively, which contain a point of $k$. Show that segments $XY$ have the same length. Find the locus of all such points $X$.

1999 North Macedonia National Olympiad, 4

Tags: combinatorial geometry , line , point

Do there exist $100$ straight lines on a plane such that they intersect each other in exactly $1999$ points?

1979 Poland - Second Round, 4

Tags: symmetry , 3d geometry , line , geometry

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 $$ S_aS_bS_cS_aS_bS_cS_bS_cS_aS_bS_cS_a = S_bS_cS_aS_bS_cS_aS_aS_bS_cS_aS_bS_c$$

2008 IMO Shortlist, 5

Tags: combinatorial geometry , point set , line , geometry

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]

2016 Singapore Junior Math Olympiad, 5

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

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 $4$)

2011 IFYM, Sozopol, 4

Tags: point , plane , line , geometry

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).

2011 Sharygin Geometry Olympiad, 18

Tags: combinatorial geometry , geometry , line , dissecting plane

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?

2000 Estonia National Olympiad, 5

Tags: combinatorics , line , combinatorial geometry

$N$ 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?

2013 Greece Team Selection Test, 4

Tags: counting , combinatorics , line , circles

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.

2000 Estonia National Olympiad, 5

Tags: combinatorial geometry , line , combinatorics

At a given plane with $2,000$ lines, all those with an odd number of different points of intersection with intersecting lines. a) Can there be an odd number of red lines if in the plane given there are no parallel lines? b) Can there be an odd number of red lines if none of any 3 given lines intersect at one point?

1967 Swedish Mathematical Competition, 1

Tags: combinatorics , rectangle , combinatorial geometry , 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?