Olympiad problems

2013 All-Russian Olympiad, 2

Tags: angle bisector , geometric transformation , circumcircle , geometry , reflection , trigonometry

Acute-angled triangle $ABC$ is inscribed into circle $\Omega$. Lines tangent to $\Omega$ at $B$ and $C$ intersect at $P$. Points $D$ and $E$ are on $AB$ and $AC$ such that $PD$ and $PE$ are perpendicular to $AB$ and $AC$ respectively. Prove that the orthocentre of triangle $ADE$ is the midpoint of $BC$.

2009 Italy TST, 2

Tags: circumcircle , reflection , geometric transformation , geometry , power of a point , radical axis , trapezoid

Two circles $O_1$ and $O_2$ intersect at $M,N$. The common tangent line nearer to $M$ of the two circles touches $O_1,O_2$ at $A,B$ respectively. Let $C,D$ be the symmetric points of $A,B$ with respect to $M$ respectively. The circumcircle of triangle $DCM$ intersects circles $O_1$ and $O_2$ at points $E,F$ respectively which are distinct from $M$. Prove that the circumradii of the triangles $MEF$ and $NEF$ are equal.

2017 Saudi Arabia BMO TST, 4

Tags: geometry , circumcenter , circumcircle , trigonometry

Let $ABC$ be a triangle with $A$ is an obtuse angle. Denote $BE$ as the internal angle bisector of triangle $ABC$ with $E \in AC$ and suppose that $\angle AEB = 45^o$. The altitude $AD$ of triangle $ABC$ intersects $BE$ at $F$. Let $O_1, O_2$ be the circumcenter of triangles $FED, EDC$. Suppose that $EO_1, EO_2$ meet $BC$ at $G, H$ respectively. Prove that $\frac{GH}{GB}= \tan \frac{a}{2}$

2014 Contests, 2

Tags: geometry , circumcircle , perpendicular bisector

Let $ABC$ be a isosceles triangle with $ AC = BC > AB$. Let $ E, F $ be the midpoints of segments $ AC, AB$, and let $l$ be the perpendicular bisector of $AC$. Let $ l $ meets $ AB$ at $K$, the line through $B$ parallel to $KC$ meets $AC$ at point $L$, and line $FL$ meets $ l$ at $W$. Let $ P $ be a point on segment $BF$. Let $H$ be the orthocenter of triangle $ACP$ and line $BH$ and $CP$ meet at point $J$. Line $FJ$ meets $l$ at $M$. Prove that $ AW = PW $ if and only if $B$ lies on the circumcircle of $EFM$.

2020 Serbian Mathematical Olympiad, Problem 4

Tags: geometry , circumcircle , trapezoid

In a trapezoid $ABCD$ such that the internal angles are not equal to $90^{\circ}$, the diagonals $AC$ and $BD$ intersect at the point $E$. Let $P$ and $Q$ be the feet of the altitudes from $A$ and $B$ to the sides $BC$ and $AD$ respectively. Circumscribed circles of the triangles $CEQ$ and $DEP$ intersect at the point $F\neq E$. Prove that the lines $AP$, $BQ$ and $EF$ are either parallel to each other, or they meet at exactly one point.

2017 Bosnia Herzegovina Team Selection Test, 6

Tags: geometry , circumcircle

Given is an acute triangle $ABC$. $M$ is an arbitrary point at the side $AB$ and $N$ is midpoint of $AC$. The foots of the perpendiculars from $A$ to $MC$ and $MN$ are points $P$ and $Q$. Prove that center of the circumcircle of triangle $PQN$ lies on the fixed line for all points $M$ from the side $AB$.

2019 Saudi Arabia JBMO TST, 1

Tags: circumcircle , parallelogram , concyclic , equal angles , geometry

Let $E$ be a point lies inside the parallelogram $ABCD$ such that $\angle BCE = \angle BAE$. Prove that the circumcenters of triangles $ABE,BCE,CDE,DAE$ are concyclic.

2021 China Team Selection Test, 5

Tags: geometry , circumcircle

Given a triangle $ABC$, a circle $\Omega$ is tangent to $AB,AC$ at $B,C,$ respectively. Point $D$ is the midpoint of $AC$, $O$ is the circumcenter of triangle $ABC$. A circle $\Gamma$ passing through $A,C$ intersects the minor arc $BC$ on $\Omega$ at $P$, and intersects $AB$ at $Q$. It is known that the midpoint $R$ of minor arc $PQ$ satisfies that $CR \perp AB$. Ray $PQ$ intersects line $AC$ at $L$, $M$ is the midpoint of $AL$, $N$ is the midpoint of $DR$, and $X$ is the projection of $M$ onto $ON$. Prove that the circumcircle of triangle $DNX$ passes through the center of $\Gamma$.

2009 German National Olympiad, 5

Tags: circumcircle , projective geometry , geometry , perpendicular bisector

Let a triangle $ ABC$. $ E,F$ in segment $ AB$ so that $ E$ lie between $ AF$ and half of circle with diameter $ EF$ is tangent with $ BC,CA$ at $ G,H$. $ HF$ cut $ GE$ at $ S$, $ HE$ cut $ FG$ at $ T$. Prove that $ C$ is midpoint of $ ST$.

2006 Turkey MO (2nd round), 1

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

Points $P$ and $Q$ on side $AB$ of a convex quadrilateral $ABCD$ are given such that $AP = BQ.$ The circumcircles of triangles $APD$ and $BQD$ meet again at $K$ and those of $APC$ and $BQC$ meet again at $L$. Show that the points $D,C,K,L$ lie on a circle.

2012 Romania National Olympiad, 2

Tags: inequalities , complex number , geometry , circumcircle

[color=darkred]Let $a$ , $b$ and $c$ be three complex numbers such that $a+b+c=0$ and $|a|=|b|=|c|=1$ . Prove that: \[3\le |z-a|+|z-b|+|z-c|\le 4,\] for any $z\in\mathbb{C}$ , $|z|\le 1\, .$[/color]

2006 Bundeswettbewerb Mathematik, 3

Tags: circumcircle , geometry

A point $P$ is given inside an acute-angled triangle $ABC$. Let $A',B',C'$ be the orthogonal projections of $P$ on sides $BC, CA, AB$ respectively. Determine the locus of points $P$ for which $\angle BAC = \angle B'A'C'$ and $\angle CBA = \angle C'B'A'$

2014 Harvard-MIT Mathematics Tournament, 7

Tags: analytic geometry , circumcircle , geometry , perpendicular bisector

Triangle $ABC$ has sides $AB = 14$, $BC = 13$, and $CA = 15$. It is inscribed in circle $\Gamma$, which has center $O$. Let $M$ be the midpoint of $AB$, let $B'$ be the point on $\Gamma$ diametrically opposite $B$, and let $X$ be the intersection of $AO$ and $MB'$. Find the length of $AX$.

2012 All-Russian Olympiad, 3

Tags: trapezoid , geometry , parallelogram , analytic geometry , circumcircle , graphing lines , slope

Consider the parallelogram $ABCD$ with obtuse angle $A$. Let $H$ be the feet of perpendicular from $A$ to the side $BC$. The median from $C$ in triangle $ABC$ meets the circumcircle of triangle $ABC$ at the point $K$. Prove that points $K,H,C,D$ lie on the same circle.

2020 Serbian Mathematical Olympiad, Problem 3

Tags: circumcircle , triangle , angle , geometry

We are given a triangle $ABC$. Points $D$ and $E$ on the line $AB$ are such that $AD=AC$ and $BE=BC$, with the arrangment of points $D - A - B - E$. The circumscribed circles of the triangles $DBC$ and $EAC$ meet again at the point $X\neq C$, and the circumscribed circles of the triangles $DEC$ and $ABC$ meet again at the point $Y\neq C$. Find the measure of $\angle ACB$ given the condition $DY+EY=2XY$.

2013 Greece Team Selection Test, 2

Tags: geometry , circumcircle

Let $ABC$ be a non-isosceles,aqute triangle with $AB<AC$ inscribed in circle $c(O,R)$.The circle $c_{1}(B,AB)$ crosses $AC$ at $K$ and $c$ at $E$. $KE$ crosses $c$ at $F$ and $BO$ crosses $KE$ at $L$ and $AC$ at $M$ while $AE$ crosses $BF$ at $D$.Prove that: i)$D,L,M,F$ are concyclic. ii)$B,D,K,M,E$ are concyclic.

2012 Sharygin Geometry Olympiad, 5

Tags: geometry , circumcircle , cyclic quadrilateral , ratio

On side $AC$ of triangle $ABC$ an arbitrary point is selected $D$. The tangent in $D$ to the circumcircle of triangle $BDC$ meets $AB$ in point $C_{1}$; point $A_{1}$ is defined similarly. Prove that $A_{1}C_{1}\parallel AC$.

2002 Kurschak Competition, 1

Tags: circumcircle , incenter , geometry

We have an acute-angled triangle which is not isosceles. We denote the orthocenter, the circumcenter and the incenter of it by $H$, $O$, $I$ respectively. Prove that if a vertex of the triangle lies on the circle $HOI$, then there must be another vertex on this circle as well.

2022 MMATHS, 10

Tags: geometry , circumcircle , circumcenter

Suppose that $A_1A_2A_3$ is a triangle with $A_1A_2 = 16$ and $A_1A_3 = A_2A_3 = 10$. For each integer $n \ge 4$, set An to be the circumcenter of triangle $A_{n-1}A_{n-2}A_{n-3}$. There exists a unique point $Z$ lying in the interiors of the circumcircles of triangles $A_kA_{k+1}A_{k+2}$ for all integers $k \ge 1$. If $ZA^2_1+ ZA^2_2+ ZA^2_3+ ZA^2_4$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd(a, b) = 1$, find $a + b$.

2009 Balkan MO Shortlist, G6

Tags: trapezoid , reflection , rotation , circumcircle , geometric transformation , trigonometry , geometry

Two circles $O_1$ and $O_2$ intersect each other at $M$ and $N$. The common tangent to two circles nearer to $M$ touch $O_1$ and $O_2$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflection of $A$ and $B$ respectively with respect to $M$. The circumcircle of the triangle $DCM$ intersect circles $O_1$ and $O_2$ respectively at points $E$ and $F$ (both distinct from $M$). Show that the circumcircles of triangles $MEF$ and $NEF$ have same radius length.

KoMaL A Problems 2022/2023, A. 844

Tags: reflection , geometry , geometric transformation , circumcircle

The inscribed circle of triangle $ABC$ is tangent to sides $BC$, $AC$ and $AB$ at points $D$, $E$ and $F$, respectively. Let $E'$ be the reflection of point $E$ across line $DF$, and $F'$ be the reflection of point $F$ across line $DE$. Let line $EF$ intersect the circumcircle of triangle $AE'F'$ at points $X$ and $Y$. Prove that $DX=DY$. [i]Proposed by Márton Lovas, Budapest[/i]

2019 Yasinsky Geometry Olympiad, p4

Tags: isosceles , geometry , circumcircle , isosceles triangle

Let $ABC$ be a triangle, $O$ is the center of the circle circumscribed around it, $AD$ the diameter of this circle. It is known that the lines $CO$ and $DB$ are parallel. Prove that the triangle $ABC$ is isosceles. (Andrey Mostovy)

2008 Iran MO (2nd Round), 3

Tags: geometry , circumcircle , trigonometry

In triangle $ABC$, $H$ is the foot of perpendicular from $A$ to $BC$. $O$ is the circumcenter of $\Delta ABC$. $T,T'$ are the feet of perpendiculars from $H$ to $AB,AC$, respectively. We know that $AC=2OT$. Prove that $AB=2OT'$.

2001 Estonia Team Selection Test, 6

Tags: geometry , circumcircle , incircle

Let $C_1$ and $C_2$ be the incircle and the circumcircle of the triangle $ABC$, respectively. Prove that, for any point $A'$ on $C_2$, there exist points $B'$ and $C'$ such that $C_1$ and $C_2$ are the incircle and the circumcircle of triangle $A'B'C'$, respectively.

2007 China National Olympiad, 1

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

Let $O, I$ be the circumcenter and incenter of triangle $ABC$. The incircle of $\triangle ABC$ touches $BC, CA, AB$ at points $D, E, F$ repsectively. $FD$ meets $CA$ at $P$, $ED$ meets $AB$ at $Q$. $M$ and $N$ are midpoints of $PE$ and $QF$ respectively. Show that $OI \perp MN$.