Found problems: 45
2000 Tournament Of Towns, 2
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 )
1974 Spain Mathematical Olympiad, 6
Two chords are drawn in a circle of radius equal to unit, $AB$ and $AC$ of equal length.
a) Describe how you can construct a third chord $DE$ that is divided into three equal parts by the intersections with $AB$ and $AC$.
b) If $AB = AC =\sqrt2$, what are the lengths of the two segments that the chord $DE$ determines in $AB$?
2013 Greece JBMO TST, 2
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 $5n$.
1986 China Team Selection Test, 4
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.
1982 Tournament Of Towns, (026) 4
(a) $10$ points dividing a circle into $10$ equal arcs are connected in pairs by $5$ chords.
Is it necessary that two of these chords are of equal length?
(b) $20$ points dividing a circle into $20$ equal arcs are connected in pairs by $10$ chords.
Prove that among these $10$ chords there are two chords of equal length.
(VV Proizvolov, Moscow)
2017 Yasinsky Geometry Olympiad, 3
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.
Kyiv City MO Juniors 2003+ geometry, 2009.89.5
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 $AED$.
1986 China Team Selection Test, 4
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.
2010 German National Olympiad, 1
Given two circles $k$ and $l$ which intersect at two points. One of their common tangents touches $k$ at point $K$, while the other common tangent touches $l$ at $L.$ Let $A$ and $B$ be the intersections of the line $KL$ with the circles $k$ and $l$, respectively. Prove that $\overline{AK} = \overline{BL}.$
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 $$ MP \cdot XY \ge 2 \cdot QX \cdot PY$$
2003 Singapore Team Selection Test, 2
Three chords $AB, CD$ and $EF$ of a circle intersect at the midpoint $M$ of $AB$. Show that if $CE$ produced and $DF$ produced meet the line $AB$ at the points $P$ and $Q$ respectively, then $M$ is also the midpoint of $PQ$.
2008 Danube Mathematical Competition, 2
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 Sharygin Geometry Olympiad, 10.3
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 $CD$.
Ukrainian TYM Qualifying - geometry, VI.9
Consider an arbitrary (optional convex) polygon. It's [i]chord [/i] is a segment whose ends lie on the boundary of the polygon, and itself belongs entirely to the polygon. Will there always be a chord of a polygon that divides it into two equal parts? Is it true that any polygon can be divided by some chord into parts, the area of each of which is not less than $\frac13$ the area of the polygon?
2011 Belarus Team Selection Test, 1
$AB$ 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
1998 Portugal MO, 5
Let $F$ be the midpoint of circle arc $AB$, and let $M$ be a point on the arc such that $AM <MB$. The perpendicular drawn from point $F$ on $AM$ intersects $AM$ at point $T$. Show that $T$ bisects the broken line $AMB$, that is $AT =TM+MB$.
KöMaL Gy. 2404. (March 1987), Archimedes of Syracuse
2005 Sharygin Geometry Olympiad, 3
Given a circle and a point $K$ inside it. An arbitrary circle equal to the given one and passing through the point $K$ has a common chord with the given circle. Find the geometric locus of the midpoints of these chords.
2014 Junior Balkan Team Selection Tests - Romania, 4
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.
1987 Austrian-Polish Competition, 1
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, 11.4
In the triangle $ABC , \angle A = \alpha, BC = a$. The inscribed circle touches the lines $AB$ and $AC$ at points $M$ and $P$. Find the length of the chord cut by the line $MP$ in a circle with diameter $BC$.
1949-56 Chisinau City MO, 31
Find the locus of the points that are the midpoints of the chords of the secant to the given circle and passing through a given point.
2018 India PRMO, 8
Let $AB$ be a chord of a circle with centre $O$. Let $C$ be a point on the circle such that $\angle ABC =30^o$ and $O$ lies inside the triangle $ABC$. Let $D$ be a point on $AB$ such that $\angle DCO = \angle OCB = 20^o$. Find the measure of $\angle CDO$ in degrees.
2018 Regional Competition For Advanced Students, 2
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]
2011 Sharygin Geometry Olympiad, 4
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.
2010 Contests, 3
Given an acute and scalene triangle $ABC$ with $AB<AC$ and random line $(e)$ that passes throuh the center of the circumscribed circles $c(O,R)$. Line $(e)$, intersects sides $BC,AC,AB$ at points $A_1,B_1,C_1$ respectively (point $C_1$ lies on the extension of $AB$ towards $B$). Perpendicular from $A$ on line $(e)$ and $AA_1$ intersect circumscribed circle $c(O,R)$ at points $M$ and $A_2$ respectively. Prove that
a) points $O,A_1,A_2, M$ are consyclic
b) if $(c_2)$ is the circumcircle of triangle $(OBC_1)$ and $(c_3)$ is the circumcircle of triangle $(OCB_1)$, then circles $(c_1),(c_2)$ and $(c_3)$ have a common chord