Found problems: 250
1935 Moscow Mathematical Olympiad, 015
Triangles $\vartriangle ABC$ and $\vartriangle A_1B_1C_1$ lie on different planes. Line $AB$ intersects line $A_1B_1$, line $BC$ intersects line $B_1C_1$ and line $CA$ intersects line $C_1A_1$. Prove that either the three lines $AA_1, BB_1, CC_1$ meet at one point or that they are all parallel.
2017 India PRMO, 26
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 $m + n + k$ ?
2014 Belarus Team Selection Test, 1
Let $AA_1, BB_1$ be the altitudes of an acute non-isosceles triangle $ABC$. Circumference of the triangles $ABC$ meets that of the triangle $A_1B_1C$ at point $N$ (different from $C$). Let $M$ be the midpoint of $AB$ and $K$ be the intersection point of $CN$ and $AB$. Prove that the line of centers the circumferences of the triangles $ABC$ and $KMC$ is parallel to the line $AB$.
(I. Kachan)
2004 Estonia Team Selection Test, 2
Let $O$ be the circumcentre of the acute triangle $ABC$ and let lines $AO$ and $BC$ intersect at point $K$. On sides $AB$ and $AC$, points $L$ and $M$ are chosen such that $|KL|= |KB|$ and $|KM| = |KC|$. Prove that segments $LM$ and $BC$ are parallel.
1999 Bosnia and Herzegovina Team Selection Test, 4
Let angle bisectors of angles $\angle BAC$ and $\angle ABC$ of triangle $ABC$ intersect sides $BC$ and $AC$ in points $D$ and $E$, respectively. Let points $F$ and $G$ be foots of perpendiculars from point $C$ on lines $AD$ and $BE$, respectively. Prove that $FG \mid \mid AB$
2004 Estonia Team Selection Test, 2
Let $O$ be the circumcentre of the acute triangle $ABC$ and let lines $AO$ and $BC$ intersect at point $K$. On sides $AB$ and $AC$, points $L$ and $M$ are chosen such that $|KL|= |KB|$ and $|KM| = |KC|$. Prove that segments $LM$ and $BC$ are parallel.
2021 Saudi Arabia BMO TST, 2
Let $ABC$ be an acute, non-isosceles triangle with $H$ the orthocenter and $M$ the midpoint of $AH$. Denote $O_1$,$O_2$ as the centers of circles pass through $H$ and respectively tangent to $BC$ at $B$, $C$. Let $X$, $Y$ be the ex-centers which respect to angle $H$ in triangles $HMO_1$,$HMO_2$. Prove that $XY$ is parallel to $O_1O_2$.
Durer Math Competition CD 1st Round - geometry, 2012.D3
Show that the planes $ACG$ and $BEH$ defined by the vertices of the cube shown in Figure are parallel. What is their distance if the edge length of the cube is $1$ meter?
[img]https://cdn.artofproblemsolving.com/attachments/c/9/21585f6c462e4289161b4a29f8805c3f63ff3e.png[/img]
2019 Dürer Math Competition (First Round), P5
Let $ABC$ be a non-right-angled triangle, with $AC\ne BC$. Let $F$ be the midpoint of side $BC$. Let $D$ be a point on line $AB$ satisfying$CA=CD$,and let $E$ be a point on line $BC$ satisfying $EB = ED$. The line passing through $A$ and parallel to $ED$ meets line $FD$ at point $I$. Line $AF$ meets line $ED$ at point $J$. Prove that points $C$, $I$ and $J$ are collinear.
2013 Thailand Mathematical Olympiad, 9
Let $ABCD$ be a convex quadrilateral, and let $M$ and$ N$ be midpoints of sides $AB$ and $CD$ respectively. Point $P$ is chosen on $CD$ so that $MP \perp CD$, and point $Q$ is chosen on $AB$ so that $NQ \perp AB$. Show that $AD \parallel BC$ if and only if $\frac{AB}{CD} =\frac{MP}{NQ}$ .
2019 India PRMO, 15
In how many ways can a pair of parallel diagonals of a regular polygon of $10$ sides be selected?
1996 Singapore Senior Math Olympiad, 1
$PQ, CD$ 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]
1973 IMO, 2
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 $M$.
2013 Dutch BxMO/EGMO TST, 1
In quadrilateral $ABCD$ the sides $AB$ and $CD$ are parallel. Let $M$ be the midpoint of diagonal $AC$. Suppose that triangles $ABM$ and $ACD$ have equal area. Prove that $DM // BC$.
2005 Oral Moscow Geometry Olympiad, 1
Given an acute-angled triangle $ABC$. A straight line parallel to $BC$ intersects sides $AB$ and $AC$ at points $M$ and $P$, respectively. At what location of the points $M$ and $P$ will the radius of the circle circumscribed about the triangle $BMP$ be the smallest?
(I. Sharygin)
2005 Sharygin Geometry Olympiad, 11.1
$A_1, B_1, C_1$ are the midpoints of the sides $BC,CA,BA$ respectively of an equilateral triangle $ABC$. Three parallel lines, passing through $A_1, B_1, C_1$ intersect, respectively, lines $B_1C_1, C_1A_1, A_1B_1$ at points $A_2, B_2, C_2$. Prove that the lines $AA_2, BB_2, CC_2$ intersect at one point lying on the circle circumscribed around the triangle $ABC$.
2016 Saudi Arabia GMO TST, 1
Let $ABC$ be an acute, non-isosceles triangle which is inscribed in a circle $(O)$. A point $I$ belongs to the segment $BC$. Denote by $H$ and $K$ the projections of $I$ on $AB$ and $AC$, respectively. Suppose that the line $HK $ intersects $(O)$ at $M, N$ ($H$ is between $M, K$ and $K$ is between $H, N$). Let $X, Y$ be the centers of the circles $(ABK),(ACH)$ respectively. Prove the
following assertions:
a) If $I$ is the projection of $A$ on $BC$, then $A$ is the center of circle $(IMN)$.
b) If $XY\parallel BC$, then the orthocenter of $XOY$ is the midpoint of $IO$.
2024 Brazil National Olympiad, 2
Let \( ABC \) be a scalene triangle. Let \( E \) and \( F \) be the midpoints of sides \( AC \) and \( AB \), respectively, and let \( D \) be any point on segment \( BC \). The circumcircles of triangles \( BDF \) and \( CDE \) intersect line \( EF \) at points \( K \neq F \), and \( L \neq E \), respectively, and intersect at points \( X \neq D \). The point \( Y \) is on line \( DX \) such that \( AY \) is parallel to \( BC \). Prove that points \( K \), \( L \), \( X \), and \( Y \) lie on the same circle.
2017 All-Russian Olympiad, 3
In the scalene triangle $ABC$,$\angle ACB=60$ and $\Omega$ is its cirumcirle.On the bisectors of the angles $BAC$ and $CBA$ points $A^\prime$,$B^\prime$ are chosen respectively such that $AB^\prime \parallel BC$ and $BA^\prime \parallel AC$.$A^\prime B^\prime$ intersects with $\Omega$ at $D,E$.Prove that triangle $CDE$ is isosceles.(A. Kuznetsov)
1992 Chile National Olympiad, 4
Given three parallel lines, prove that there are three points, one on each line, which are the vertices of an equilateral triangle.
2014 Switzerland - Final Round, 1
The points $A, B, C$ and $D$ lie in this order on the circle $k$. Let $t$ be the tangent at $k$ through $C$ and $s$ the reflection of $AB$ at $AC$. Let $G$ be the intersection of the straight line $AC$ and $BD$ and $H$ the intersection of the straight lines $s$ and $CD$. Show that $GH$ is parallel to $t$.
2016 Oral Moscow Geometry Olympiad, 2
A regular heptagon $A_1A_2A_3A_4A_5A_6A_7$ is given. Straight $A_2A_3$ and $A_5A_6$ intersect at point $X$, and straight lines $A_3A_5$ and $A_1A_6$ intersect at point $Y$. Prove that lines $A_1A_2$ and $XY$ are parallel.
Kyiv City MO Juniors Round2 2010+ geometry, 2012.9.4
In an acute-angled triangle $ABC$, the point $O$ is the center of the circumcircle, and the point $H$ is the orthocenter. It is known that the lines $OH$ and $BC$ are parallel, and $BC = 4OH $. Find the value of the smallest angle of triangle $ ABC $.
(Black Maxim)
2018 Junior Balkan Team Selection Tests - Romania, 2
Let $ABC$ be an acute triangle, with $AB \ne AC$. Let $D$ be the midpoint of the line segment $BC$, and let $E$ and $F$ be the projections of $D$ onto the sides $AB$ and $AC$, respectively. If $M$ is the midpoint of the line segment $EF$, and $O$ is the circumcenter of triangle $ABC$, prove that the lines $DM$ and $AO$ are parallel.
[hide=PS] As source was given [url=https://artofproblemsolving.com/community/c629086_caucasus_mathematical_olympiad]Caucasus MO[/url], but I was unable to find this problem in the contest collections [/hide]
Indonesia Regional MO OSP SMA - geometry, 2018.3
Let $ \Gamma_1$ and $\Gamma_2$ be two different circles with the radius of same length and centers at points $O_1$ and $O_2$, respectively. Circles $\Gamma_1$ and $\Gamma_2$ are tangent at point $P$. The line $\ell$ passing through $O_1$ is tangent to $\Gamma_2$ at point $A$. The line $\ell$ intersects $\Gamma_1$ at point $X$ with $X$ between $A$ and $O_1$. Let $M$ be the midpoint of $AX$ and $Y$ the intersection of $PM$ and $\Gamma_2$ with $Y\ne P$. Prove that $XY$ is parallel to $O_1O_2$.