Found problems: 75
1975 Czech and Slovak Olympiad III A, 4
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 $xy.$
2002 Estonia National Olympiad, 4
Find the maximum length of a broken line on the surface of a unit cube, such that its links are the cube’s edges and diagonals of faces, the line does not intersect itself and passes no more than once through any vertex of the cube, and its endpoints are in two opposite vertices of the cube.
2021 Durer Math Competition (First Round), 5
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.
2001 239 Open Mathematical Olympiad, 6
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.
2018 Thailand TSTST, 2
$9$ horizontal and $9$ vertical lines are drawn through a square. Prove that it is possible to select $20$ rectangles so that the sides of each rectangle is a segment of one of the given lines (including the sides of the square), and for any two of the $20$ rectangles, it is possible to cover one of them with the other (rotations are allowed).
2018 Dutch IMO TST, 1
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.
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 $XB$.
2017 EGMO, 3
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?
1959 Putnam, B1
Let each of $m$ distinct points on the positive part of the $x$-axis be joined to $n$ distinct points on the positive part of the $y$-axis. Obtain a formula for the number of intersection points of these segments, assuming that no three of the segments are concurrent.
1999 North Macedonia National Olympiad, 4
Do there exist $100$ straight lines on a plane such that they intersect each other in exactly $1999$ points?
1998 German National Olympiad, 1
Find all possible numbers of lines in a plane which intersect in exactly $37$ points.
2000 Estonia National Olympiad, 5
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?
1976 Czech and Slovak Olympiad III A, 6
Consider two non-parallel half-planes $\pi,\pi'$ with the common boundary line $p.$ Four different points $A,B,C,D$ are given in the half-plane $\pi.$ Similarly, four points $A',B',C',D'\in\pi'$ are given such that $AA'\parallel BB'\parallel CC'\parallel DD'$. Moreover, none of these points lie on $p$ and the points $A,B,C,D'$ form a tetrahedron. Show that the points $A',B',C',D$ also form a tetrahedron with the same volume as $ABCD'.$
2009 Germany Team Selection Test, 2
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]
2014 Junior Balkan Team Selection Tests - Romania, 3
Let $ABC$ be an acute triangle and $D \in (BC) , E \in (AD)$ be mobile points. The circumcircle of triangle $CDE$ meets the median from $C$ of the triangle $ABC$ at $F$ Prove that the circumcenter of triangle $AEF$ lies on a fixed line.
2016 IMAR Test, 2
Given a positive integer $n$, does there exist a planar polygon and a point in its plane such that every line through that point meets the boundary of the polygon at exactly $2n$ points?
1962 Czech and Slovak Olympiad III A, 3
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 $X$.
Durer Math Competition CD Finals - geometry, 2017.C2
The triangle $ABC$ is isosceles and has a right angle at the vertex $A$. Construct all points that simultaneously satisfy the following two conditions:
(i) are equidistant from points $A$ and $B$
(ii) heve distance exactly three times from point $C$ as far as from point $B$.
2016 Singapore Senior Math Olympiad, 2
Let $n$ be a positive integer. Determine the minimum number of lines that can be drawn on the plane so that they intersect in exactly $n$ distinct points.
2022 Turkey EGMO TST, 5
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.
1986 All Soviet Union Mathematical Olympiad, 428
A line is drawn through the $A$ vertex of triangle $ABC$ with $|AB|\ne|AC|$. Prove that the line can not contain more than one point $M$ such, that $M$ is not a triangle vertex, and $\angle ABM = \angle ACM$. What lines do not contain such a point $M$ at all?
1978 Bundeswettbewerb Mathematik, 2
A set of $n^2$ counters are labeled with $1,2,\ldots, n$, each label appearing $n$ times. Can one arrange the counters on a line in such a way that for all $x \in \{1,2,\ldots, n\}$, between any two successive counters with the label $x$ there are exactly $x$ counters (with labels different from $x$)?
1973 Putnam, A6
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.
1989 Tournament Of Towns, (221) 5
We are given $N$ lines ($N > 1$ ) in a plane, no two of which are parallel and no three of which have a point in common. Prove that it is possible to assign, to each region of the plane determined by these lines, a non-zero integer of absolute value not exceeding $N$ , such that the sum of the integers o n either side of any of the given lines is equal to $0$ .
(S . Fomin, Leningrad)
2019 Tournament Of Towns, 2
$2019$ 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)