Olympiad problems

2018 JBMO Shortlist, G6

Let $XY$ be a chord of a circle $\Omega$, with center $O$, which is not a diameter. Let $P, Q$ be two distinct points inside the segment $XY$, where $Q$ lies between $P$ and $X$. Let $\ell$ the perpendicular line drawn from $P$ to the diameter which passes through $Q$. Let $M$ be the intersection point of $\ell$ and $\Omega$, which is closer to $P$. Prove that $$ MP \cdot XY \ge 2 \cdot QX \cdot PY$$

1986 China Team Selection Test, 4

Tags: combinatorics unsolved , combinatorics , points , Chords , China , TST , Team Selection Test

Mark $4 \cdot k$ points in a circle and number them arbitrarily with numbers from $1$ to $4 \cdot k$. The chords cannot share common endpoints, also, the endpoints of these chords should be among the $4 \cdot k$ points. [b]i.[/b] Prove that $2 \cdot k$ pairwisely non-intersecting chords can be drawn for each of whom its endpoints differ in at most $3 \cdot k - 1$. [b]ii.[/b] Prove that the $3 \cdot k - 1$ cannot be improved.

2008 Danube Mathematical Competition, 2

Tags: geometry , Chords , midpoint , circles , concurrency , concurrent

In a triangle $ABC$ let $A_1$ be the midpoint of side $BC$. Draw circles with centers $A, A1$ and radii $AA_1, BC$ respectively and let $A'A''$ be their common chord. Similarly denote the segments $B'B''$ and $C'C''$. Show that lines $A'A'', B'B'''$ and $C'C''$ are concurrent.

2005 Paraguay Mathematical Olympiad, 5

Tags: geometry , equal segments , Chords

Given a chord $PQ$ of a circle and $M$ the midpoint of the chord, let $AB$ and $CD$ be two chords that pass through $M$. $AC$ and $BD$ are drawn until $PQ$ is intersected at points $X$ and $Y$ respectively. Show that $X$ and $Y$ are equidistant from $M$.

1999 Portugal MO, 3

Tags: geometry , Chords

If two parallel chords of a circumference, $10$ mm and $14$ mm long, with distance $6$ mm from each other, how long is the chord equidistant from these two?

2017 India PRMO, 26

Tags: geometry , Chords , parallel , area , circle

Let $AB$ and $CD$ be two parallel chords in a circle with radius $5$ such that the centre $O$ lies between these chords. Suppose $AB = 6, CD = 8$. Suppose further that the area of the part of the circle lying between the chords $AB$ and $CD$ is $(m\pi + n) / k$, where $m, n, k$ are positive integers with gcd$(m, n, k) = 1$. What is the value of $m + n + k$ ?

2018 Regional Competition For Advanced Students, 2

Tags: geometry , Equilateral Triangle , perpendicular bisector , Chords , Austria , national olympiad , AUT

Let $k$ be a circle with radius $r$ and $AB$ a chord of $k$ such that $AB > r$. Furthermore, let $S$ be the point on the chord $AB$ satisfying $AS = r$. The perpendicular bisector of $BS$ intersects $k$ in the points $C$ and $D$. The line through $D$ and $S$ intersects $k$ for a second time in point $E$. Show that the triangle $CSE$ is equilateral. [i]Proposed by Stefan Leopoldseder[/i]

2013 Greece JBMO TST, 2

Tags: combinatorics , combinatorial geometry , circle , Chords , concurrent

Consider $n$ different points lying on a circle, such that there are not three chords defined by that point that pass through the same interior point of the circle. a) Find the value of $n$, if the numbers of triangles that are defined using $3$ of the n points is equal to $2n$ b) Find the value of $n$, if the numbers of the intersection points of the chords that are interior to the circle is equal to $5n$.

Kyiv City MO Juniors 2003+ geometry, 2009.89.5

Tags: geometry , Chords , orthocenter , circle

A chord $AB$ is drawn in the circle, on which the point $P$ is selected in such a way that $AP = 2PB$. The chord $DE$ is perpendicular to the chord $AB $ and passes through the point $P$. Prove that the midpoint of the segment $AP$ is the orthocener of the triangle $AED$.

2018 Finnish National High School Mathematics Comp, 3

Tags: geometry , Chords , midpoint , circle

The chords $AB$ and $CD$ of a circle intersect at $M$, which is the midpoint of the chord $PQ$. The points $X$ and $Y$ are the intersections of the segments $AD$ and $PQ$, respectively, and $BC$ and $PQ$, respectively. Show that $M$ is the midpoint of $XY$.

2017 Yasinsky Geometry Olympiad, 3

Tags: geometry , Chords , angle bisector

In a circle, let $AB$ and $BC$ be chords , with $AB =\sqrt3, BC =3\sqrt3, \angle ABC =60^o$. Find the length of the circle chord that divides angle $ \angle ABC$ in half.

1987 Austrian-Polish Competition, 1

Tags: geometry , 3D geometry , sphere , Chords

Three pairwise orthogonal chords of a sphere $S$ are drawn through a given point $P$ inside $S$. Prove that the sum of the squares of their lengths does not depend on their directions.

2005 Sharygin Geometry Olympiad, 1

Tags: geometry , Chords , perpendicular

The chords $AC$ and $BD$ of the circle intersect at point $P$. The perpendiculars to $AC$ and $BD$ at points $C$ and $D$, respectively, intersect at point $Q$. Prove that the lines $AB$ and $PQ$ are perpendicular.

2011 Sharygin Geometry Olympiad, 21

Tags: geometry , Chords , midpoints , arc midpoint , arc , concurrency , concurrent

On a circle with diameter $AC$, let $B$ be an arbitrary point distinct from $A$ and $C$. Points $M, N$ are the midpoints of chords $AB, BC$, and points $P, Q$ are the midpoints of smaller arcs restricted by these chords. Lines $AQ$ and $BC$ meet at point $K$, and lines $CP$ and $AB$ meet at point $L$. Prove that lines $MQ, NP$ and $KL$ concur.

1993 Mexico National Olympiad, 5

Tags: geometry , collinear , Chords

$OA, OB, OC$ are three chords of a circle. The circles with diameters $OA, OB$ meet again at $Z$, the circles with diameters $OB, OC$ meet again at $X$, and the circles with diameters $OC, OA$ meet again at $Y$. Show that $X, Y, Z$ are collinear.

1986 China Team Selection Test, 4

Tags: combinatorics unsolved , combinatorics , points , Chords , China , TST , Team Selection Test

Mark $4 \cdot k$ points in a circle and number them arbitrarily with numbers from $1$ to $4 \cdot k$. The chords cannot share common endpoints, also, the endpoints of these chords should be among the $4 \cdot k$ points. [b]i.[/b] Prove that $2 \cdot k$ pairwisely non-intersecting chords can be drawn for each of whom its endpoints differ in at most $3 \cdot k - 1$. [b]ii.[/b] Prove that the $3 \cdot k - 1$ cannot be improved.

2014 Junior Balkan Team Selection Tests - Romania, 4

Tags: geometry , Concyclic , Chords , midpoints

In a circle, consider two chords $[AB], [CD]$ that intersect at $E$, lines $AC$ and $BD$ meet at $F$. Let $G$ be the projection of $E$ onto $AC$. We denote by $M,N,K$ the midpoints of the segment lines $[EF] ,[EA]$ and $[AD]$, respectively. Prove that the points $M, N,K,G$ are concyclic.

2019 Tournament Of Towns, 2

Tags: orthocenter , geometry , Chords , equal segments

Let $ABC$ be an acute triangle. Suppose the points $A',B',C'$ lie on its sides $BC,AC,AB$ respectively and the segments $AA',BB',CC'$ intersect in a common point $P$ inside the triangle. For each of those segments let us consider the circle such that the segment is its diameter, and the chord of this circle that contains the point $P$ and is perpendicular to this diameter. All three these chords occurred to have the same length. Prove that $P$ is the orthocenter of the triangle $ABC$. (Grigory Galperin)

2000 Tournament Of Towns, 2

Tags: geometry , Chords , equal segments , Circumcenter , circumcircle

The chords $AC$ and $BD$ of a, circle with centre $O$ intersect at the point $K$. The circumcentres of triangles $AKB$ and $CKD$ are $M$ and $N$ respectively. Prove that $OM = KN$. (A Zaslavsky )

2011 Belarus Team Selection Test, 1

Tags: conics , parabola , circles , Chords , parallel , geometry

$AB$ and $CD$ are two parallel chords of a parabola. Circle $S_1$ passing through points $A,B$ intersects circle $S_2$ passing through $C,D$ at points $E,F$. Prove that if $E$ belongs to the parabola, then $F$ also belongs to the parabola. I.Voronovich

2014 Abels Math Contest (Norwegian MO) Final, 3b

Tags: combinatorics , Coloring , combinatorial geometry , Chords

Nine points are placed on a circle. Show that it is possible to colour the $36$ chords connecting them using four colours so that for any set of four points, each of the four colours is used for at least one of the six chords connecting the given points

2011 Sharygin Geometry Olympiad, 4

Tags: circle , geometry , regular polygon , hexagon , inscribed , Chords , combinatorics

Given the circle of radius $1$ and several its chords with the sum of lengths $1$. Prove that one can be inscribe a regular hexagon into that circle so that its sides don’t intersect those chords.

2005 Sharygin Geometry Olympiad, 10.3

Tags: circles , Chords , parallel , distance , geometry

Two parallel chords $AB$ and $CD$ are drawn in a circle with center $O$. Circles with diameters $AB$ and $CD$ intersect at point $P$. Prove that the midpoint of the segment $OP$ is equidistant from lines $AB$ and $CD$.

1994 Italy TST, 1

Tags: min , max , geometry , perpendicular , Chords , circle

Given a circle $\gamma$ and a point $P$ inside it, find the maximum and minimum value of the sum of the lengths of two perpendicular chords of $\gamma$ passing through $P$.

1996 Singapore Senior Math Olympiad, 1

Tags: bisects segment , Chords , geometry , parallel , Tangents , circle

$PQ, CD$ are parallel chords of a circle. The tangent at $D$ cuts $PQ$ at $T$ and $B$ is the point of contact of the other tangent from $T$ (Fig. ). Prove that $BC$ bisects $PQ$. [img]https://cdn.artofproblemsolving.com/attachments/2/f/22f69c03601fbb8e388e319cd93567246b705c.png[/img]