This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 75

2016 Singapore Junior Math Olympiad, 5
Tags: combinatorics , combinatorial geometry , line , min , point
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 $4$)

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.

1955 Moscow Mathematical Olympiad, 295
Tags: combinatorial geometry , geometry , convex , line
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.

2019 Tournament Of Towns, 2
Tags: game , game strategy , combinatorics , line
$2019$ 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)

1973 Putnam, A6
Tags: line , geometry
Prove that it is impossible for seven distinct straight lines to be situated in the euclidean plane so as to have at least six points where exactly three of these lines intersect and at least four points where exactly two of these lines intersect.

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)

1961 Putnam, B2
Tags: probability , line
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 $a?$

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 $a$?

1997 Estonia National Olympiad, 4
Tags: combinatorics , line , combinatorial geometry
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.

2021 Durer Math Competition (First Round), 5
Tags: combinatorial geometry , line , 3d geometry , geometry , plane , combinatorics
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.

1975 Czech and Slovak Olympiad III A, 4
Tags: parameterization , algebra , linear algebra , absolute value , analytic geometry , line
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 $xy.$

2000 Estonia National Olympiad, 5
Tags: line , combinatorial geometry , 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?

1987 Tournament Of Towns, (153) 4
Tags: bisects area , line construction , geometry , arc , broken line , line
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.

2011 Sharygin Geometry Olympiad, 18
Tags: geometry , combinatorial 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?

II Soros Olympiad 1995 - 96 (Russia), 10.7
Tags: inradius , line , geometry
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]

2011 IFYM, Sozopol, 4
Tags: plane , line , point , 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).

1953 Moscow Mathematical Olympiad, 252
Tags: angle , concurrency , 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.

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 $$ S_aS_bS_cS_aS_bS_cS_bS_cS_aS_bS_cS_a = S_bS_cS_aS_bS_cS_aS_aS_bS_cS_aS_bS_c$$

1948 Moscow Mathematical Olympiad, 147
Tags: combinatorial geometry , line , geometry
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]

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.

EGMO 2017, 3
Tags: plane , combinatorics , combinatorial geometry , line
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?

Kvant 2019, M2558
Tags: game , game strategy , combinatorics , line
$2019$ 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)

2022 Turkey EGMO TST, 5
Tags: geometry , line , circles , tangent
We are given three points $A,B,C$ on a semicircle. The tangent lines at $A$ and $B$ to the semicircle meet the extension of the diameter at points $M,N$ respectively. The line passing through $A$ that is perpendicular to the diameter meets $NC$ at $R$, and the line passing through $B$ that is perpendicular to the diameter meets $MC$ at $S$. If the line $RS$ meets the extension of the diameter at $Z$, prove that $ZC$ is tangent to the semicircle.

2008 IMO Shortlist, 5
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]

2009 QEDMO 6th, 2
Tags: combinatorics , coloring , line
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.)