Olympiad problems

2008 Tournament Of Towns, 2

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.

1999 Argentina National Olympiad, 2

Tags: geometry , circles , tangent , segment

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 $QQ'$.

2005 Sharygin Geometry Olympiad, 13

Tags: geometry , minimum , segment , construction

A triangle $ABC$ and two lines $\ell_1, \ell_2$ are given. Through an arbitrary point $D$ on the side $AB$, a line parallel to $\ell_1$ intersects the $AC$ at point $E$ and a line parallel to $\ell_2$ intersects the $BC$ at point $F$. Construct a point $D$ for which the segment $EF$ has the smallest length.

1984 Brazil National Olympiad, 4

Tags: right triangle , minimum , projection , segment , geometry

$ABC$ is a triangle with $\angle A = 90^o$. For a point $D$ on the side $BC$, the feet of the perpendiculars to $AB$ and $AC$ are $E$ and$ F$. For which point $D$ is $ EF$ a minimum?

2009 Junior Balkan Team Selection Tests - Romania, 3

Tags: segment , geometry , concurrency

Consider a regular polygon $A_0A_1...A_{n-1}, n \ge 3$, and $m \in\{1, 2, ..., n - 1\}, m \ne n/2$. For any number $i \in \{0,1, ... , n - 1\}$, let $r(i)$ be the remainder of $i + m$ at the division by $n$. Prove that no three segments $A_iA_{r(i)}$ are concurrent.

2016 Oral Moscow Geometry Olympiad, 6

Tags: paper , segment , geometry , folding , square

Given a square sheet of paper with a side of $2016$. Is it possible to bend its not more than ten times, construct a segment of length $1$?

Cono Sur Shortlist - geometry, 1993.12

Tags: geometry , segment

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.

2011 Argentina National Olympiad, 6

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

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.

1981 Czech and Slovak Olympiad III A, 2

Tags: combinatorial geometry , geometry , closed , segment

Let $n$ be a positive integer. Consider $n^2+1$ (closed, i.e. including endpoints) segments on a single line. Show that at least one of the following statements holds: a) there are $n+1$ segments with non-empty intersection, b) there are $n+1$ segments among which two of them are disjoint.

2012 Czech-Polish-Slovak Junior Match, 4

Tags: geometry , rhombus , segment

A rhombus $ABCD$ is given with $\angle BAD = 60^o$ . Point $P$ lies inside the rhombus such that $BP = 1$, $DP = 2$, $CP = 3$. Determine the length of the segment $AP$.

2019 Auckland Mathematical Olympiad, 2

Tags: geometry , segment

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.

1946 Moscow Mathematical Olympiad, 119

Tags: minimum , geometry , segment

On the legs of $\angle AOB$, the segments $OA$ and $OB$ lie, $OA > OB$. Points $M$ and $N$ on lines $OA$ and $OB$, respectively, are such that $AM = BN = x$. Find $x$ for which the length of $MN$ is minimal.

1990 All Soviet Union Mathematical Olympiad, 515

Tags: segment , product , geometry

The point $P$ lies inside the triangle $ABC$. A line is drawn through $P$ parallel to each side of the triangle. The lines divide $AB$ into three parts length $c, c', c"$ (in that order), and $BC$ into three parts length $a, a', a"$ (in that order), and $CA$ into three parts length $b, b', b"$ (in that order). Show that $abc = a'b'c' = a"b"c"$.

2017 Hanoi Open Mathematics Competitions, 12

Tags: fixed , geometry , segment

Let $(O)$ denote a circle with a chord $AB$, and let $W$ be the midpoint of the minor arc $AB$. Let $C$ stand for an arbitrary point on the major arc $AB$. The tangent to the circle $(O)$ at $C$ meets the tangents at $A$ and $B$ at points $X$ and $Y$, respectively. The lines $W X$ and $W Y$ meet $AB$ at points $N$ and $M$ , respectively. Does the length of segment $NM$ depend on position of $C$ ?

2020 Canadian Junior Mathematical Olympiad, 2

Tags: combinatorics , segment

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?

2009 Greece Team Selection Test, 4

Tags: combinatorics , coloring , minimization , point set , segment

Given are $N$ points on the plane such that no three of them are collinear,which are coloured red,green and black.We consider all the segments between these points and give to each segment a [i]"value"[/i] according to the following conditions: [b]i.[/b]If at least one of the endpoints of a segment is black then the segment's [i]"value"[/i] is $0$. [b]ii.[/b]If the endpoints of the segment have the same colour,re or green,then the segment's [i]"value"[/i] is $1$. [b]iii.[/b]If the endpoints of the segment have different colours but none of them is black,then the segment's [i]"value"[/i] is $-1$. Determine the minimum possible sum of the [i]"values"[/i] of the segments.

2006 German National Olympiad, 4

Tags: triangle , distance , segment , geometry

Let $D$ be a point inside a triangle $ABC$ such that $|AC| -|AD| \geq 1$ and $|BC|- |BD| \geq 1.$ Prove that for any point $E$ on the segment $AB$, we have $|EC| -|ED| \geq 1.$

2009 Greece Team Selection Test, 4

Tags: combinatorics , coloring , minimization , point set , segment

Given are $N$ points on the plane such that no three of them are collinear,which are coloured red,green and black.We consider all the segments between these points and give to each segment a [i]"value"[/i] according to the following conditions: [b]i.[/b]If at least one of the endpoints of a segment is black then the segment's [i]"value"[/i] is $0$. [b]ii.[/b]If the endpoints of the segment have the same colour,re or green,then the segment's [i]"value"[/i] is $1$. [b]iii.[/b]If the endpoints of the segment have different colours but none of them is black,then the segment's [i]"value"[/i] is $-1$. Determine the minimum possible sum of the [i]"values"[/i] of the segments.

2007 Rioplatense Mathematical Olympiad, Level 3, 5

Tags: combinatorial geometry , triangle , segment , combinatorics

Divide each side of a triangle into $50$ equal parts, and each point of the division is joined to the opposite vertex by a segment. Calculate the number of intersection points determined by these segments. Clarification : the vertices of the original triangle are not considered points of intersection or division.

2020 Tournament Of Towns, 1

Tags: parabola , segment , analytic geometry , algebra

Consider two parabolas $y = x^2$ and $y = x^2 - 1$. Let $U$ be the set of points between the parabolas (including the points on the parabolas themselves). Does $U$ contain a line segment of length greater than $10^6$ ? Alexey Tolpygo

2000 Switzerland Team Selection Test, 9

Tags: intersecting circles , geometric construction , construction , geometry , segment , max

Two given circles $k_1$ and $k_2$ intersect at points $P$ and $Q$. Construct a segment $AB$ through $P$ with the endpoints at $k_1$ and $k_2$ for which $AP \cdot PB$ is maximal.

2016 Peru Cono Sur TST, P6

Tags: circles , fixed , segment , geometry

Two circles $\omega_1$ and $\omega_2$, which have centers $O_1$ and $O_2$, respectively, intersect at $A$ and $B$. A line $\ell$ that passes through $B$ cuts to $\omega_1$ again at $C$ and cuts to $\omega_2$ again in $D$, so that points $C, B, D$ appear in that order. The tangents of $\omega_1$ and $\omega_2$ in $C$ and $D$, respectively, intersect in $E$. Line $AE$ intersects again to the circumscribed circumference of the triangle $AO_1O_2$ in $F$. Try that the length of the $EF$ segment is constant, that is, it does not depend on the choice of $\ell$.

2014 Sharygin Geometry Olympiad, 2

Tags: geometry , tangent , constant , segment

A circle, its chord $AB$ and the midpoint $W$ of the minor arc $AB$ are given. Take an arbitrary point $C$ on the major arc $AB$. The tangent to the circle at $C$ meets the tangents at $A$ and $B$ at points $X$ and $Y$ respectively. Lines $WX$ and WY meet AB at points $N$ and $M$ respectively. Prove that the length of segment $NM$ does not depend on point $C$. (A. Zertsalov, D. Skrobot)

1983 All Soviet Union Mathematical Olympiad, 369

Tags: segment , combinatorial geometry , geometry

The $M$ set consists of $k$ non-intersecting segments on the line. It is possible to put an arbitrary segment shorter than $1$ cm on the line in such a way, that his ends will belong to $M$. Prove that the total sum of the segment lengths is not less than $1/k$ cm.

1997 Poland - Second Round, 3

Tags: combinatorics , combinatorial geometry , coloring , segment

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