Olympiad problems

2007 Iran Team Selection Test, 1

In triangle $ABC$, $M$ is midpoint of $AC$, and $D$ is a point on $BC$ such that $DB=DM$. We know that $2BC^{2}-AC^{2}=AB.AC$. Prove that \[BD.DC=\frac{AC^{2}.AB}{2(AB+AC)}\]

2017 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: geometry , angle bisector , circumcircle

It is given triangle $ABC$. Let internal and external angle bisector of angle $\angle BAC$ intersect line $BC$ in points $D$ and $E$, respectively, and circumcircle of triangle $ADE$ intersects line $AC$ in point $F$. Prove that $FD$ is angle bisector of $\angle BFC$

2007 Postal Coaching, 4

Tags: geometry , circumradius , circumcircle , angle bisector

Let $BE$ and $CF$ be the bisectors of $\angle B$ and $\angle C$ of a triangle $ABC$ whose incentre is $I$. Suppose $EF$, extended, meets the circumcircle of $ABC$ in $M,N$. Show that the circumradius of $MIN$ is twice that of $ABC$.

2010 Contests, 2

Tags: circumcircle , ratio , geometric transformation , reflection , geometry

In an acute-angled triangle $ABC$, $M$ is the midpoint of side $BC$, and $D, E$ and $F$ the feet of the altitudes from $A, B$ and $C$, respectively. Let $H$ be the orthocenter of $\Delta ABC$, $S$ the midpoint of $AH$, and $G$ the intersection of $FE$ and $AH$. If $N$ is the intersection of the median $AM$ and the circumcircle of $\Delta BCH$, prove that $\angle HMA = \angle GNS$. [i]Proposed by Marko Djikic[/i]

2014 Contests, 1

Tags: circumcircle , projective geometry , geometry

Let $ABC$ an acute triangle and $\Gamma$ its circumcircle. The bisector of $BAC$ intersects $\Gamma$ at $M\neq A$. A line $r$ parallel to $BC$ intersects $AC$ at $X$ and $AB$ at $Y$. Also, $MX$ and $MY$ intersect $\Gamma$ again at $S$ and $T$, respectively. If $XY$ and $ST$ intersect at $P$, prove that $PA$ is tangent to $\Gamma$.

2021 Saint Petersburg Mathematical Olympiad, 3

Tags: geometry , 3d geometry , pyramid , circumcircle

In the pyramid $SA_1A_2 \cdots A_n$, all sides are equal. Let point $X_i$ be the midpoint of arc $A_iA_{i+1}$ in the circumcircle of $\triangle SA_iA_{i+1}$ for $1 \le i \le n$ with indices taken mod $n$. Prove that the circumcircles of $X_1A_2X_2, X_2A_3X_3, \cdots, X_nA_1X_1$ have a common point.

2022 Thailand Online MO, 8

Tags: geometry , circumcircle

Let $ABCD$ be a convex quadrilateral with $AD = BC$, $\angle BAC+\angle DCA = 180^{\circ}$, and $\angle BAC \neq 90^{\circ}.$ Let $O_1$ and $O_2$ be the circumcenters of triangles $ABC$ and $CAD$, respectively. Prove that one intersection point of the circumcircles of triangles $O_1BC$ and $O_2AD$ lies on $AC$.

2024 Polish MO Finals, 6

Tags: right angle , parallelogram , circumcircle , geometry

Let $ABCD$ be a parallelogram. Let $X \notin AC $ lie inside $ABCD$ so that $\angle AXB = \angle CXD = 90^ {\circ}$ and $\Omega$ denote the circumcircle of $AXC$. Consider a diameter $EF$ of $\Omega$ and assume neither $E, \ X, \ B$ nor $F, \ X, \ D$ are collinear. Let $K \neq X$ be an intersection point of circumcircles of $BXE$ and $DXF$ and $L \neq X$ be such point on $\Omega$ so that $\angle KXL = 90^{\circ}$. Prove that $AB = KL$.

2003 France Team Selection Test, 1

Tags: circumcircle , homothety , incenter , geometry , tangent circle

Let $B$ be a point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of $AC$ from $B$. Prove that the circumcentre of triangle $BCD$ lies on the circumcircle of triangle $ABC$.

Swiss NMO - geometry, 2016.1

Tags: geometry , equal angles , circumcircle , angle

Let $ABC$ be a triangle with $\angle BAC = 60^o$. Let $E$ be the point on the side $BC$ , such that $2 \angle BAE = \angle ACB$ . Let $D$ be the second intersection of $AB$ and the circumcircle of the triangle $AEC$ and $P$ be the second intersection of $CD$ and the circumcircle of the triangle $DBE$. Calculate the angle $\angle BAP$.

2015 Indonesia MO Shortlist, G3

Tags: tangent , geometry , circumcircle , incircle

Given $ABC$ triangle with incircle $L_1$ and circumcircle $L_2$. If points $X, Y, Z$ lie on $L_2$, such that $XY, XZ$ are tangent to $L_1$, then prove that $YZ$ is also tangent to $L_1$.

1987 IMO Longlists, 70

Tags: geometry , circumcircle , trigonometry , area of a triangle

In an acute-angled triangle $ABC$ the interior bisector of angle $A$ meets $BC$ at $L$ and meets the circumcircle of $ABC$ again at $N$. From $L$ perpendiculars are drawn to $AB$ and $AC$, with feet $K$ and $M$ respectively. Prove that the quadrilateral $AKNM$ and the triangle $ABC$ have equal areas.[i](IMO Problem 2)[/i] [i]Proposed by Soviet Union.[/i]

Estonia Open Senior - geometry, 2004.2.4

Tags: circumcircle , right angle , geometry

On the circumcircle of triangle $ABC$, point $P$ is chosen, such that the perpendicular drawn from point $P$ to line $AC$ intersects the circle again at a point $Q$, the perpendicular drawn from point $Q$ to line $AB$ intersects the circle again at a point $R$ and the perpendicular drawn from point $R$ to line $BC$ intersects the circle again at the initial point $P$. Let $O$ be the centre of this circle. Prove that $\angle POC = 90^o$.

2013 Iran Team Selection Test, 17

Tags: asymptote , circumcircle , angle bisector , perpendicular bisector , geometry

In triangle $ABC$, $AD$ and $AH$ are the angle bisector and the altitude of vertex $A$, respectively. The perpendicular bisector of $AD$, intersects the semicircles with diameters $AB$ and $AC$ which are drawn outside triangle $ABC$ in $X$ and $Y$, respectively. Prove that the quadrilateral $XYDH$ is concyclic. [i]Proposed by Mahan Malihi[/i]

2024 Argentina Cono Sur TST, 3

Tags: geometry , circumcircle

Let $ABC$ be an acute triangle. The point $B'$ of the line $CA$ is such that $A$, $C$ and $B'$ are in that order on the line and $B'C=AB$; the point $C'$ of the line $AB$ is such that $A$, $B$ and $C'$ are in that order on the line and $C'B=AC$. Prove that the circumcenter of triangle $AB'C'$ belongs to the circumcircle of triangle $ABC$.

2006 Kyiv Mathematical Festival, 4

Tags: circumcircle , incenter , trigonometry , trig identities , law of sines , geometry

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $O$ be the circumcenter and $H$ be the intersection point of the altitudes of acute triangle $ABC.$ The straight lines $BH$ and $CH$ intersect the segments $CO$ and $BO$ at points $D$ and $E$ respectively. Prove that if triangles $ODH$ and $OEH$ are isosceles then triangle $ABC$ is isosceles too.

2010 Vietnam Team Selection Test, 2

Tags: circumcircle , power of a point , radical axis , geometry

Let $ABC$ be a triangle with $ \widehat{BAC}\neq 90^\circ $. Let $M$ be the midpoint of $BC$. We choose a variable point $D$ on $AM$. Let $(O_1)$ and $(O_2)$ be two circle pass through $ D$ and tangent to $BC$ at $B$ and $C$. The line $BA$ and $CA$ intersect $(O_1),(O_2)$ at $ P,Q$ respectively. [b]a)[/b] Prove that tangent line at $P$ on $(O_1)$ and $Q$ on $(O_2)$ must intersect at $S$. [b]b)[/b] Prove that $S$ lies on a fix line.

2016 Sharygin Geometry Olympiad, 1

Tags: geometry , angle chasing , parallelogram , circumcircle

The diagonals of a parallelogram $ABCD$ meet at point $O$. The tangent to the circumcircle of triangle $BOC$ at $O$ meets ray $CB$ at point $F$. The circumcircle of triangle $FOD$ meets $BC$ for the second time at point $G$. Prove that $AG=AB$.

2003 Italy TST, 2

Tags: circumcircle , geometry

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 $ABC$.

2021 ITAMO, 5

Tags: geometry , circumcircle

Let $ABC$ be an acute-angled triangle, let $M$ be the midpoint of $BC$ and let $H$ be the foot of the $B$-altitude. Let $Q$ be the circumcenter of $ABM$ and let $X$ be the intersection point between $BH$ and the axis of $BC$. Show that the circumcircles of the two triangles $ACM$, $AXH$ and the line $CQ$ pass through a same point if and only if $BQ$ is perpendicular to $CQ$.

2020 Ukraine Team Selection Test, 3

Tags: circumcircle , tangent line , angle chasing , trigonometry , geometry

Altitudes $AH1$ and $BH2$ of acute triangle $ABC$ intersect at $H$. Let $w1$ be the circle that goes through $H2$ and touches the line $BC$ at $H1$, and let $w2$ be the circle that goes through $H1$ and touches the line $AC$ at $H2$. Prove, that the intersection point of two other tangent lines $BX$ and $AY$( $X$ and $Y$ are different from $H1$ and $H2$) to circles $w1$ and $w2$ respectively, lies on the circumcircle of triangle $HXY$. Proposed by [i]Danilo Khilko[/i]

2018 China Western Mathematical Olympiad, 5

Tags: geometry , circumcircle

In acute triangle $ABC,$ $AB<AC,$ $O$ is the circumcenter of the triangle. $M$ is the midpoint of segment $BC,$ $(AOM)$ intersects the line $AB$ again at $D$ and intersects the segment $AC$ at $E.$ Prove that $DM=EC.$

1978 IMO, 1

Tags: geometry , inradius , circumcircle , incenter , triangle

In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$

2021 Dutch IMO TST, 3

Tags: geometry , orthocenter , collinear , circumcircle

Let $ABC$ be an acute-angled and non-isosceles triangle with orthocenter $H$. Let $O$ be the center of the circumscribed circle of triangle $ABC$ and let $K$ be center of the circumscribed circle of triangle $AHO$. Prove that the reflection of $K$ wrt $OH$ lies on $BC$.

1956 Moscow Mathematical Olympiad, 333

Tags: altitude , circumcenter , circumcircle , concurrency , geometry

Let $O$ be the center of the circle circumscribed around $\vartriangle ABC$, let $A_1, B_1, C_1$ be symmetric to $O$ through respective sides of $\vartriangle ABC$. Prove that all altitudes of $\vartriangle A_1B_1C_1$ pass through $O$, and all altitudes of $\vartriangle ABC$ pass through the center of the circle circumscribed around $\vartriangle A_1B_1C_1$.