Olympiad problems

2005 Switzerland - Final Round, 1

Tags: median , geometry , equilateral , centroid , concyclic , cyclic quadrilateral

Let $ABC$ be any triangle and $D, E, F$ the midpoints of $BC, CA, AB$. The medians $AD, BE$ and $CF$ intersect at point $S$. At least two of the quadrilaterals $AF SE, BDSF, CESD$ are cyclic. Show that the triangle $ABC$ is equilateral.

1995 Poland - Second Round, 5

Tags: geometry , 3d geometry , tetrahedron , concyclic , incircle

The incircles of the faces $ABC$ and $ABD$ of a tetrahedron $ABCD$ are tangent to the edge $AB$ in the same point. Prove that the points of tangency of these incircles to the edges $AC,BC,AD,BD$ are concyclic.

2010 Bundeswettbewerb Mathematik, 3

Tags: projection , concyclic , collinear , altitude , perpendicular , geometry

Given an acute-angled triangle $ABC$. Let $CB$ be the altitude and $E$ a random point on the line $CD$. Finally, let $P, Q, R$ and $S$ are the projections of $D$ on the straight lines $AC, AE, BE$ and $BC$. Prove that the points $P, Q, R$ and $S$ lie either on a circle or on one straight line.

2000 Switzerland Team Selection Test, 13

Tags: geometry , incircle , concyclic

The incircle of a triangle $ABC$ touches the sides $AB,BC,CA$ at points $D,E,F$ respectively. Let $P$ be an internal point of triangle $ABC$ such that the incircle of triangle $ABP$ touches $AB$ at $D$ and the sides $AP$ and $BP$ at $Q$ and $R$. Prove that the points $E,F,R,Q$ lie on a circle.

Kyiv City MO Seniors 2003+ geometry, 2017.11.5

Tags: geometry , concyclic , isosceles , orthocenter

In the acute isosceles triangle $ABC$ the altitudes $BB_1$ and $CC_1$ are drawn, which intersect at the point $H$. Let $L_1$ and $L_2$ be the feet of the angle bisectors of the triangles $B_1AC_1$ and $B_1HC_1$ drawn from vertices $A$ and $H$, respectively. The circumscribed circles of triangles $AHL_1$ and $AHL_2$ intersects the line $B_1C_1$ for the second time at points $P$ and $Q$, respectively. Prove that points $B, C, P$ and $Q$ lie on the same circle. (M. Plotnikov, D. Hilko)

1989 Czech And Slovak Olympiad IIIA, 1

Tags: geometry , concyclic

Three different points $A, B, C $ lying on a circle with center $S$ and a line $p$ perpendicular to $ AS$ are given in the plane. Let's mark the intersections of the line $p$ with the lines $AB$, $AC$ as $D$ and $E$. Prove that the points $B, C, D, E$ lie on the same circle.

2019 Saudi Arabia BMO TST, 2

Tags: geometry , incenter , excenter , concyclic

Let $I $be the incenter of triangle $ABC$and $J$ the excenter of the side $BC$: Let $M$ be the midpoint of $CB$ and $N$ the midpoint of arc $BAC$ of circle $(ABC)$. If $T$ is the symmetric of the point $N$ by the point $A$, prove that the quadrilateral $JMIT$ is cyclic

2022 Grand Duchy of Lithuania, 3

Tags: geometry , concyclic , isosceles

The center $O$ of the circle $\omega$ passing through the vertex $C$ of the isosceles triangle $ABC$ ($AB = AC$) is the interior point of the triangle $ABC$. This circle intersects segments $BC$ and $AC$ at points $D \ne C$ and $E \ne C$, respectively, and the circumscribed circle $\Omega$ of the triangle $AEO$ at the point $F \ne E$. Prove that the center of the circumcircle of the triangle $BDF$ lies on the circle $\Omega$.

2006 Estonia Team Selection Test, 2

Tags: projection , circumcenter , geometry , concyclic , cyclic , independent

The center of the circumcircle of the acute triangle $ABC$ is $O$. The line $AO$ intersects $BC$ at $D$. On the sides $AB$ and $AC$ of the triangle, choose points $E$ and $F$, respectively, so that the points $A, E, D, F$ lie on the same circle. Let $E'$ and $F'$ projections of points $E$ and $F$ on side $BC$ respectively. Prove that length of the segment $E'F'$ does not depend on the position of points $E$ and $F$.

2000 Singapore MO Open, 1

Tags: geometry , circumcircle , perpendicular , concyclic

Triangle $ABC$ is inscribed in a circle with center $O$. Let $D$ and $E$ be points on the respective sides $AB$ and $AC$ so that $DE$ is perpendicular to $AO$. Show that the four points $B,D,E$ and $C$ lie on a circle.

2018 Ecuador NMO (OMEC), 5

Tags: cyclic quadrilateral , concyclic , equal segments

Let $ABC$ be an acute triangle and let $M$, $N$, and $ P$ be on $CB$, $AC$, and $AB$, respectively, such that $AB = AN + PB$, $BC = BP + MC$, $CA = CM + AN$. Let $\ell$ be a line in a different half plane than $C$ with respect to to the line $AB$ such that if $X, Y$ are the projections of $A, B$ on $\ell$ respectively, $AX = AP$ and $BY = BP$. Prove that $NXYM$ is a cyclic quadrilateral.

1999 All-Russian Olympiad Regional Round, 9.2

Tags: geometry , concyclic

In triangle $ABC$, on side $AC$ there are points $D$ and $E$, that $AB = AD$ and $BE = EC$ ($E$ between $A$ and $D$). Point $F$ is midpoint of arc $BC$ of circumcircle of triangle $ABC$. Prove that the points $B, E, D, F$ lie on the same circle.

Geometry Mathley 2011-12, 15.2

Tags: geometry , concyclic

Let $O$ be the centre of the circumcircle of triangle $ABC$. Point $D$ is on the side $BC$. Let $(K)$ be the circumcircle of $ABD$. $(K)$ meets $AO$ at $E$ that is distinct from $A$. (a) Prove that $B,K,O,E$ are on the same circle that is called $(L)$. (b) $(L)$ intersects $AB$ at $F$ distinct $B$. Point $G$ is on $(L)$ such that $EG \parallel OF$. $GK$ meets $AD$ at $S, SO$ meets $BC$ at $T$ . Prove that $O,E, T,C$ are on the same circle. Trần Quang Hùng

2001 Korea Junior Math Olympiad, 8

Tags: geometry , concyclic

$ABCD$ is a convex quadrilateral, both $\angle ABC$ and $\angle BCD$ acute. $E$ is a point inside $ABCD$ satisfying $AE=DE$, and $X, Y$ are the intersection of $AD$ and $CE, BE$ respectively, but not $X=A$ or $Y=D$. If $ABEX$ and $CDEY$ are both inscribed quadrilaterals, prove that the distance between $E$ and the lines $AB, BC, CD$ are all equal.

2006 Sharygin Geometry Olympiad, 10.2

Tags: projection , symmetry , concyclic , geometry

The projections of the point $X$ onto the sides of the $ABCD$ quadrangle lie on the same circle. $Y$ is a point symmetric to $X$ with respect to the center of this circle. Prove that the projections of the point $B$ onto the lines $AX,XC, CY, YA$ also lie on the same circle.

Estonia Open Senior - geometry, 2013.2.3

Tags: geometry , concyclic , circles

Circles $c_1, c_2$ with centers $O_1, O_2$, respectively, intersect at points $P$ and $Q$ and touch circle c internally at points $A_1$ and $A_2$, respectively. Line $PQ$ intersects circle c at points $B$ and $D$. Lines $A_1B$ and $A_1D$ intersect circle $c_1$ the second time at points $E_1$ and $F_1$, respectively, and lines $A_2B$ and $A_2D$ intersect circle $c_2$ the second time at points $ E_2$ and $F_2$, respectively. Prove that $E_1, E_2, F_1, F_2$ lie on a circle whose center coincides with the midpoint of line segment $O_1O_2$.

2015 India PRMO, 16

Tags: angle chasing , geometry , concyclic

$16.$ In an acute angle triangle $ABC,$ let $D$ be the foot of the altitude from $A,$ and $E$ be the midpoint of $BC.$ Let $F$ be the midpoint of $AC.$ Suppose $\angle{BAE}=40^o. $ If $\angle{DAE}=\angle{DFE},$ What is the magnitude of $\angle{ADF}$ in degrees $?$

2018 Puerto Rico Team Selection Test, 2

Tags: geometry , cyclic quadrilateral , equal angles , concyclic

Let $ABC$ be an acute triangle and let $P,Q$ be points on $BC$ such that $\angle QAC =\angle ABC$ and $\angle PAB = \angle ACB$. We extend $AP$ to $M$ so that $ P$ is the midpoint of $AM$ and we extend $AQ$ to $N$ so that $Q$ is the midpoint of $AN$. If T is the intersection point of $BM$ and $CN$, show that quadrilateral $ABTC$ is cyclic.

2016 Germany Team Selection Test, 1

Tags: concyclic , geometry , circles

The two circles $\Gamma_1$ and $\Gamma_2$ with the midpoints $O_1$ resp. $O_2$ intersect in the two distinct points $A$ and $B$. A line through $A$ meets $\Gamma_1$ in $C \neq A$ and $\Gamma_2$ in $D \neq A$. The lines $CO_1$ and $DO_2$ intersect in $X$. Prove that the four points $O_1,O_2,B$ and $X$ are concyclic.

2018 Adygea Teachers' Geometry Olympiad, 3

Tags: geometry , concyclic , circles

Two circles intersect at points $A$ and $B$. Through point $B$, a straight line intersects the circles at points $C$ and $D$, and then tangents to the circles are drawn through points $C$ and $D$. Prove that the points $A, D, C$ and $P$ - the intersection point of the tangents - lie on the same circle.

Indonesia Regional MO OSP SMA - geometry, 2019.5

Tags: geometry , circumcircle , angle bisector , concyclic

Given triangle $ABC$, with $AC> BC$, and the it's circumcircle centered at $O$. Let $M$ be the point on the circumcircle of triangle $ABC$ so that $CM$ is the bisector of $\angle ACB$. Let $\Gamma$ be a circle with diameter $CM$. The bisector of $BOC$ and bisector of $AOC$ intersect $\Gamma$ at $P$ and $Q$, respectively. If $K$ is the midpoint of $CM$, prove that $P, Q, O, K$ lie at one point of the circle.

Croatia MO (HMO) - geometry, 2012.7

Tags: geometry , cyclic quadrilateral , concyclic , incircle , right triangle , excircle

Let the points $M$ and $N$ be the intersections of the inscribed circle of the right-angled triangle $ABC$, with sides $AB$ and $CA$ respectively , and points $P$ and $Q$ respectively be the intersections of the ex-scribed circles opposite to vertices $B$ and $C$ with direction $BC$. Prove that the quadrilateral $MNPQ$ is a cyclic if and only if the triangle $ABC$ is right-angled with a right angle at the vertex $A$.

2019 Philippine TST, 4

Tags: geometry , reflection , concyclic , parallelogram

Let $P$ be a point in parallelogram $ABCD$ such that $$PA \cdot PC + PB \cdot PD = AB \cdot BC.$$ Prove that the reflections of $P$ over lines $AB$, $BC$, $CD$, and $DA$ are concyclic.

1995 Czech and Slovak Match, 5

Tags: diagonal , convex quadrilateral , reflection , concyclic , orthogonal , geometry

The diagonals of a convex quadrilateral $ABCD$ are orthogonal and intersect at point $E$. Prove that the reflections of $E$ in the sides of quadrilateral $ABCD$ lie on a circle.

Swiss NMO - geometry, 2006.5

Tags: geometry , concyclic , circles

A circle $k_1$ lies within a second circle $k_2$ and touches it at point $A$. A line through $A$ intersects $k_1$ again in $B$ and $k_2$ in $C$. The tangent to $k_1$ through $B$ intersects $k_2$ at points $D$ and $E$. The tangents at $k_1$ passing through $C$ intersects $k_1$ in points $F$ and $G$. Prove that $D, E, F$ and $G$ lie on a circle.