Olympiad problems

1973 IMO Shortlist, 6

Establish if there exists a finite set $M$ of points in space, not all situated in the same plane, so that for any straight line $d$ which contains at least two points from M there exists another straight line $d'$, parallel with $d,$ but distinct from $d$, which also contains at least two points from $M$.

2018 IFYM, Sozopol, 6

Tags: straight lines , intersection , floor function

There are $a$ straight lines in a plane, no two of which are parallel to each other and no three intersect in one point. a) Prove that there exist a straight line for which each of the two Half-Planes defined by it contains at least $\lfloor \frac{(a-1)(a-2)}{10} \rfloor$ intersection points. b) Find all $a$ for which the evaluation in a) is the best possible.

1986 IMO Shortlist, 11

Tags: combinatorial geometry , point set , straight lines , geometry

Let $f(n)$ be the least number of distinct points in the plane such that for each $k = 1, 2, \cdots, n$ there exists a straight line containing exactly $k$ of these points. Find an explicit expression for $f(n).$ [i]Simplified version.[/i] Show that $f(n)=\left[\frac{n+1}{2}\right]\left[\frac{n+2}{2}\right].$ Where $[x]$ denoting the greatest integer not exceeding $x.$

1973 IMO, 2

Tags: combinatorics , point set , 3d geometry , straight lines , parallel

Establish if there exists a finite set $M$ of points in space, not all situated in the same plane, so that for any straight line $d$ which contains at least two points from M there exists another straight line $d'$, parallel with $d,$ but distinct from $d$, which also contains at least two points from $M$.

1992 IMO Longlists, 15

Tags: counting , intersection , straight lines , combinatorics

Prove that there exist $78$ lines in the plane such that they have exactly $1992$ points of intersection.

2008 India Regional Mathematical Olympiad, 1

Tags: geometry , trigonometry , semicircle , straight lines , intersection

On a semicircle with diameter $AB$ and centre $S$, points $C$ and $D$ are given such that point $C$ belongs to arc $AD$. Suppose $\angle CSD = 120^\circ$. Let $E$ be the point of intersection of the straight lines $AC$ and $BD$ and $F$ the point of intersection of the straight lines $AD$ and $BC$. Prove that $EF=\sqrt{3}AB$.

1986 IMO Longlists, 7

Tags: geometry , point set , combinatorial geometry , straight lines

Let $f(n)$ be the least number of distinct points in the plane such that for each $k = 1, 2, \cdots, n$ there exists a straight line containing exactly $k$ of these points. Find an explicit expression for $f(n).$ [i]Simplified version.[/i] Show that $f(n)=\left[\frac{n+1}{2}\right]\left[\frac{n+2}{2}\right].$ Where $[x]$ denoting the greatest integer not exceeding $x.$