Found problems: 3882
2003 Italy TST, 2
Let $B\not= A$ be a point on the tangent to circle $S_1$ through the point $A$ on the circle. A point $C$ outside the circle is chosen so that segment $AC$ intersects the circle in two distinct points. Let $S_2$ be the circle tangent to $AC$ at $C$ and to $S_1$ at some point $D$, where $D$ and $B$ are on the opposite sides of the line $AC$. Let $O$ be the circumcentre of triangle $BCD$. Show that $O$ lies on the circumcircle of triangle $ABC$.
2011 Junior Balkan Team Selection Tests - Romania, 2
Let $ ABC$ be a triangle with circumcentre $ O$. The points $ P$ and $ Q$ are interior points of the sides $ CA$ and $ AB$ respectively. Let $ K,L$ and $ M$ be the midpoints of the segments $ BP,CQ$ and $ PQ$. respectively, and let $ \Gamma$ be the circle passing through $ K,L$ and $ M$. Suppose that the line $ PQ$ is tangent to the circle $ \Gamma$. Prove that $ OP \equal{} OQ.$
[i]Proposed by Sergei Berlov, Russia [/i]
2005 Sharygin Geometry Olympiad, 5
There are two parallel lines $p_1$ and $p_2$. Points $A$ and $B$ lie on $p_1$, and $C$ on $p_2$. We will move the segment $BC$ parallel to itself and consider all the triangles $AB'C '$ thus obtained. Find the locus of the points in these triangles:
a) points of intersection of heights,
b) the intersection points of the medians,
c) the centers of the circumscribed circles.
2011 Tuymaada Olympiad, 2
A circle passing through the vertices $A$ and $B$ of a cyclic quadrilateral $ABCD$ intersects diagonals $AC$ and $BD$ at $E$ and $F$, respectively. The lines $AF$ and $BC$ meet at a point $P$, and the lines $BE$ and $AD$ meet at a point $Q$. Prove that $PQ$ is parallel to $CD$.
1998 Vietnam National Olympiad, 2
Let be given a tetrahedron whose circumcenter is $O$. Draw diameters $AA_{1},BB_{1},CC_{1},DD_{1}$ of the circumsphere of $ABCD$. Let $A_{0},B_{0},C_{0},D_{0}$ be the centroids of triangle $BCD,CDA,DAB,ABC$. Prove that $A_{0}A_{1},B_{0}B_{1},C_{0}C_{1},D_{0}D_{1}$ are concurrent at a point, say, $F$. Prove that the line through $F$ and a midpoint of a side of $ABCD$ is perpendicular to the opposite side.
2018 Czech and Slovak Olympiad III A, 3
In triangle $ABC$ let be $D$ an intersection of $BC$ and the $A$-angle bisector. Denote $E,F$ the circumcenters of $ABD$ and $ACD$ respectively. Assuming that the circumcenter of $AEF$ lies on the line $BC$ what is the possible size of the angle $BAC$ ?
2022 Israel TST, 3
In triangle $ABC$, the angle bisectors are $BE$ and $CF$ (where $E, F$ are on the sides of the triangle), and their intersection point is $I$. Point $N$ lies on the circumcircle of $AEF$, and the angle $\angle IAN$ is right. The circumcircle of $AEF$ meets the line $NI$ a second time at the point $L$. Show that the circumcenter of $AIL$ lies on line $BC$.
2014 ELMO Shortlist, 2
$ABCD$ is a cyclic quadrilateral inscribed in the circle $\omega$. Let $AB \cap CD = E$, $AD \cap BC = F$. Let $\omega_1, \omega_2$ be the circumcircles of $AEF, CEF$, respectively. Let $\omega \cap \omega_1 = G$, $\omega \cap \omega_2 = H$. Show that $AC, BD, GH$ are concurrent.
[i]Proposed by Yang Liu[/i]
2012 Sharygin Geometry Olympiad, 6
Let $ABC$ be an isosceles triangle with $BC = a$ and $AB = AC = b$. Segment $AC$ is the base of an isosceles triangle $ADC$ with $AD = DC = a$ such that points $D$ and $B$ share the opposite sides of AC. Let $CM$ and $CN$ be the bisectors in triangles $ABC$ and $ADC$ respectively. Determine the circumradius of triangle $CMN$.
(M.Rozhkova)
1990 USAMO, 5
An acute-angled triangle $ABC$ is given in the plane. The circle with diameter $\, AB \,$ intersects altitude $\, CC' \,$ and its extension at points $\, M \,$ and $\, N \,$, and the circle with diameter $\, AC \,$ intersects altitude $\, BB' \,$ and its extensions at $\, P \,$ and $\, Q \,$. Prove that the points $\, M, N, P, Q \,$ lie on a common circle.
2015 Swedish Mathematical Competition, 1
Given the acute triangle $ABC$. A diameter of the circumscribed circle of the triangle intersects the sides $AC$ and $BC$, dividing the side $BC$ in half. Show that the same diameter divides the side $AC$ in a ratio of $1: 3$, calculated from $A$, if and only if $\tan B = 2 \tan C$.
2001 German National Olympiad, 6 (12)
Let $ABC$ be a triangle with $\angle A = 90^o$ and $\angle B < \angle C$. The tangent at $A$ to the circumcircle $k$ of $\vartriangle ABC$ intersects line $BC$ at $D$. Let $E$ be the reflection of $A$ in $BC$. Also, let $X$ be the feet of the perpendicular from $A$ to $BE$ and let $Y$ be the midpoint of $AX$. Line $BY$ meets $k$ again at $Z$. Prove that line $BD$ is tangent to the circumcircle of $\vartriangle ADZ$.
2015 Junior Balkan Team Selection Tests - Romania, 1
Let $ABC$ be an acute triangle with $AB \neq AC$ . Also let $M$ be the midpoint of the side $BC$ , $H$ the orthocenter of the triangle $ABC$ , $O_1$ the midpoint of the segment $AH$ and $O_2$ the center of the circumscribed circle of the triangle $BCH$ . Prove that $O_1AMO_2$ is a parallelogram .
2019 Latvia Baltic Way TST, 10
Let $\triangle ABC$ be an acute angled triangle with orthocenter $H$ and let $M$ be a midpoint of $BC$. Circle with diameter $AH$ is $\omega_1$ and circle with center $M$ is $\omega_2$. If $\omega_2$ is tangent to circumcircle of $\triangle ABC$, then prove that circles $\omega_1$ and $\omega_2$ are tangent to each other.
1999 Balkan MO, 1
Let $O$ be the circumcenter of the triangle $ABC$. The segment $XY$ is the diameter of the circumcircle perpendicular to $BC$ and it meets $BC$ at $M$. The point $X$ is closer to $M$ than $Y$ and $Z$ is the point on $MY$ such that $MZ = MX$. The point $W$ is the midpoint of $AZ$.
a) Show that $W$ lies on the circle through the midpoints of the sides of $ABC$;
b) Show that $MW$ is perpendicular to $AY$.
2019 China Team Selection Test, 5
In $\Delta ABC$, $AD \perp BC$ at $D$. $E,F$ lie on line $AB$, such that $BD=BE=BF$. Let $I,J$ be the incenter and $A$-excenter. Prove that there exist two points $P,Q$ on the circumcircle of $\Delta ABC$ , such that $PB=QC$, and $\Delta PEI \sim \Delta QFJ$ .
2016 Hong Kong TST, 3
Let $ABC$ be a triangle such that $AB \neq AC$. The incircle with centre $I$ touches $BC$ at $D$. Line $AI$ intersects the circumcircle $\Gamma$ of $ABC$ at $M$, and $DM$ again meets $\Gamma$ at $P$. Find $\angle API$
2003 Baltic Way, 12
Points $M$ and $N$ are taken on the sides $BC$ and $CD$ respectively of a square $ABCD$ so that $\angle MAN=45^{\circ}$. Prove that the circumcentre of $\triangle AMN$ lies on $AC$.
2003 IMAR Test, 3
The exinscribed circle of a triangle $ABC$ corresponding to its vertex $A$ touches the sidelines $AB$ and $AC$ in the points $M$ and $P$, respectively, and touches its side $BC$ in the point $N$. Show that if the midpoint of the segment $MP$ lies on the circumcircle of triangle $ABC$, then the points $O$, $N$, $I$ are collinear, where $I$ is the incenter and $O$ is the circumcenter of triangle $ABC$.
2018 Federal Competition For Advanced Students, P2, 4
Let $ABC$ be a triangle and $P$ a point inside the triangle such that the centers $M_B$ and $M_A$ of the circumcircles $k_B$ and $k_A$ of triangles $ACP$ and $BCP$, respectively, lie outside the triangle $ABC$. In addition, we assume that the three points $A, P$ and $M_A$ are collinear as well as the three points $B, P$ and $M_B$. The line through $P$ parallel to side $AB$ intersects circles $k_A$ and $k_B$ in points $D$ and $E$, respectively, where $D, E \ne P$. Show that $DE = AC + BC$.
[i](Proposed by Walther Janous)[/i]
2016 Croatia Team Selection Test, Problem 3
Let $P$ be a point inside a triangle $ABC$ such that
$$ \frac{AP + BP}{AB} = \frac{BP + CP}{BC} = \frac{CP + AP}{CA} .$$
Lines $AP$, $BP$, $CP$ intersect the circumcircle of triangle $ABC$ again in $A'$, $B'$, $C'$. Prove that the triangles $ABC$ and $A'B'C'$ have a common incircle.
2016 Saint Petersburg Mathematical Olympiad, 5
Points $A$ and $P$ are marked in the plane not lying on the line $\ell$. For all right triangles $ABC$ with hypotenuse on $\ell$, show that the circumcircle of triangle $BPC$ passes through a fixed point other than $P$.
2002 All-Russian Olympiad, 3
Let O be the circumcenter of a triangle ABC. Points M and N are choosen on the sides AB and BC respectively so that the angle AOC is two times greater than angle MON. Prove that the perimeter of triangle MBN is not less than the lenght of side AC
2010 IMO Shortlist, 3
Let $A_1A_2 \ldots A_n$ be a convex polygon. Point $P$ inside this polygon is chosen so that its projections $P_1, \ldots , P_n$ onto lines $A_1A_2, \ldots , A_nA_1$ respectively lie on the sides of the polygon. Prove that for arbitrary points $X_1, \ldots , X_n$ on sides $A_1A_2, \ldots , A_nA_1$ respectively,
\[\max \left\{ \frac{X_1X_2}{P_1P_2}, \ldots, \frac{X_nX_1}{P_nP_1} \right\} \geq 1.\]
[i]Proposed by Nairi Sedrakyan, Armenia[/i]
2018 China Northern MO, 1
In triangle $ABC$, let the circumcenter, incenter, and orthocenter be $O$, $I$, and $H$ respectively. Segments $AO$, $AI$, and $AH$ intersect the circumcircle of triangle $ABC$ at $D$, $E$, and $F$. $CD$ intersects $AE$ at $M$ and $CE$ intersects $AF$ at $N$. Prove that $MN$ is parallel to $BC$.