Found problems: 59
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.
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?
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).
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 $XB$.
2014 Chile National Olympiad, 6
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]
2008 IMO Shortlist, 5
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]
1949-56 Chisinau City MO, 58
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?
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.
2023 4th Memorial "Aleksandar Blazhevski-Cane", P1
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]