Olympiad problems

2019 Auckland Mathematical Olympiad, 2

Tags: geometry , segments

There are $2019$ segments $[a_1, b_1]$, $...$, $[a_{2019}, b_{2019}]$ on the line. It is known that any two of them intersect. Prove that they all have a point in common.

2008 Tournament Of Towns, 2

Tags: ratio , combinatorial geometry , combinatorics , segments

There are ten congruent segments on a plane. Each intersection point divides every segment passing through it in the ratio $3:4$. Find the maximum number of intersection points.

2022 Belarusian National Olympiad, 10.2

Tags: combinatorics , segments

A positive integer $n$ is given. On the segment $[0,n]$ of the real line $m$ distinct segments whose endpoints have integer coordinates are chosen. It turned out that it is impossible to choose some of thos segments such that their total length is $n$ and their union is $[0,n]$ Find the maximum possible value of $m$

2011 Greece JBMO TST, 2

Tags: combinatorics , combinatorial geometry , segments , intersections

On every side of a square $ABCD$, we consider three points different (to each other). a) Find the number of line segments defined with endpoints those points , that do not lie on sides of the square. b) If there are no three of the previous line segments passing through the same point, find how many of the intersection points of those segmens line in the interior of the square.

1997 Poland - Second Round, 3

Tags: combinatorics , combinatorial geometry , Coloring , segments

Let be given $n$ points, no three of which are on a line. All the segments with endpoints in these points are colored so that two segments with a common endpoint are of different colors. Determine the least number of colors for which this is possible

Cono Sur Shortlist - geometry, 1993.12

Tags: geometry , segments

Given $4$ lines in the plane such that there are not $2$ parallel to each other or no $3$ concurrent, we consider the following $ 8$ segments: in each line we have $2$ consecutive segments determined by the intersections with the other three lines. Prove that: a) The lengths of the $ 8$ segments cannot be the numbers $1, 2, 3,4, 5, 6, 7, 8$ in some order. b) The lengths of the $ 8$ segments can be $ 8$ different integers.

1999 Argentina National Olympiad, 2

Tags: geometry , segments , circles , Tangents

Let $C_1$ and $C_2$ be the outer circumferences of centers $O_1$ and $O_2$, respectively. The two tangents to the circumference $C_2$ are drawn by $O_1$, intersecting $C_1$ at $P$ and $P'$. The two tangents to the circumference $C_1$ are drawn by $O_2$, intersecting $C_2$ at $Q$ and $Q'$. Prove that the segment $PP'$ is equal to the segment $QQ'$.

2011 Argentina National Olympiad, 6

Tags: geometry , combinatorics , game , segments , square , game strategy , winning strategy

We have a square of side $1$ and a number $\ell$ such that $0 <\ell <\sqrt2$. Two players $A$ and $B$, in turn, draw in the square an open segment (without its two ends) of length $\ell $, starts A. Each segment after the first cannot have points in common with the previously drawn segments. He loses the player who cannot make his play. Determine if either player has a winning strategy.

1984 Tournament Of Towns, (072) 3

Tags: segments , combinatorial geometry , points , geometry

On a plane there is a finite set of $M$ points, no three of which are collinear . Some points are joined to others by line segments, with each point connected to no more than one line segment . If we have a pair of intersecting line segments $AB$ and $CD$ we decide to replace them with $AC$ and $BD$, which are opposite sides of quadrilateral $ABCD$. In the resulting system of segments we decide to perform a similar substitution, if possible, and so on . Is it possible that such substitutions can be carried out indefinitely? (V.E. Kolosov)

2020 Canadian Junior Mathematical Olympiad, 2

Tags: combinatorics , segments

Ziquan makes a drawing in the plane for art class. He starts by placing his pen at the origin, and draws a series of line segments, such that the $n^{th}$ line segment has length $n$. He is not allowed to lift his pen, so that the end of the $n^{th}$ segment is the start of the $(n + 1)^{th}$ segment. Line segments drawn are allowed to intersect and even overlap previously drawn segments. After drawing a finite number of line segments, Ziquan stops and hands in his drawing to his art teacher. He passes the course if the drawing he hands in is an $N$ by $N$ square, for some positive integer $N$, and he fails the course otherwise. Is it possible for Ziquan to pass the course?