Olympiad problems

1995 Singapore Team Selection Test, 2

Let $ABC$ be an acute-angled triangle. Suppose that the altitude of $\vartriangle ABC$ at $B$ intersects the circle with diameter $AC$ at $P$ and $Q$, and the altitude at $C$ intersects the circle with diameter $AB$ at $M$ and $N$. Prove that $P, Q, M$ and $N$ lie on a circle.

2017 Ukrainian Geometry Olympiad, 4

Tags: circumcircle , incircle , midline , concyclic , geometry

In the right triangle $ABC$ with hypotenuse $AB$, the incircle touches $BC$ and $AC$ at points ${{A}_{1}}$ and ${{B}_{1}}$ respectively. The straight line containing the midline of $\Delta ABC$ , parallel to $AB$, intersects its circumcircle at points $P$ and $T$. Prove that points $P,T,{{A}_{1}}$ and ${{B}_{1}}$ lie on one circle.

2022 Korea Junior Math Olympiad, 6

Tags: circumcircle , concyclic , tangent , geometry

Let $ABC$ be a isosceles triangle with $\overline{AB}=\overline{AC}$. Let $D(\neq A, C)$ be a point on the side $AC$, and circle $\Omega$ is tangent to $BD$ at point $E$, and $AC$ at point $C$. Denote by $F(\neq E)$ the intersection of the line $AE$ and the circle $\Omega$, and $G(\neq a)$ the intersection of the line $AC$ and the circumcircle of the triangle $ABF$. Prove that points $D, E, F,$ and $G$ are concyclic.

Ukrainian TYM Qualifying - geometry, 2020.13

Tags: concyclic , excenter , geometry

In the triangle $ABC$ on the side $BC$, the points$ D$ and $E$ are chosen so that the angle $BAD$ is equal to the angle $EAC$. Let $I$ and $J$ be the centers of the inscribed circles of triangles $ABD$ and $AEC$ respectively, $F$ be the point of intersection of $BI$ and $EJ$, $G$ be the point of intersection of $DI$ and $CJ$. Prove that the points $I, J, F, G$ lie on one circle, the center of which belongs to the line $I_bI_c$, where $I_b$ and $I_c$ are the centers of the exscribed circles of the triangle $ABC$, which touch respectively sides $AC$ and $AB$.

2017 Greece National Olympiad, 1

Tags: triangle , concyclic , geometry , cyclic quadrilateral

An acute triangle $ABC$ with $AB<AC<BC$ is inscribed in a circle $c(O,R)$. The circle $c_1(A,AC)$ intersects the circle $c$ at point $D$ and intersects $CB$ at $E$. If the line $AE$ intersects $c$ at $F$ and $G$ lies in $BC$ such that $EB=BG$, prove that $F,E,D,G$ are concyclic.

1979 Chisinau City MO, 178

Tags: concyclic , geometry

Prove that the bases of the altitudes and medians of an acute-angled triangle lie on the same circle.

1967 Poland - Second Round, 6

Tags: geometry , concyclic , 3d geometry , sphere

Prove that the points $ A_1, A_2, \ldots, A_n $ ($ n \geq 7 $) located on the surface of the sphere lie on a circle if and only if the planes tangent to the surface of the sphere at these points have a common point or are parallel to one straight line.

2020 Kosovo National Mathematical Olympiad, 3

Tags: concyclic , circumcenter , circumcircle , geometry

Let $\triangle ABC$ be a triangle. Let $O$ be the circumcenter of triangle $\triangle ABC$ and $P$ a variable point in line segment $BC$. The circle with center $P$ and radius $PA$ intersects the circumcircle of triangle $\triangle ABC$ again at another point $R$ and $RP$ intersects the circumcircle of triangle $\triangle ABC$ again at another point $Q$. Show that points $A$, $O$, $P$ and $Q$ are concyclic.

2010 All-Russian Olympiad Regional Round, 10.3

Tags: geometry , concyclic , angle bisector

In triangle $ABC$, the angle bisectors $AD$, $BE$ and $CF$ are drawn, intersecting at point $I$. The perpendicular bisector of the segment $AD$ intersects lines $BE$ and $CF$ at points $M$ and $N$, respectively. Prove that points $A$, $I$, $M$ and $ N$ lie on the same circle.

Novosibirsk Oral Geo Oly IX, 2021.2

Tags: concyclic , geometry

The robot crawls the meter in a straight line, puts a flag on and turns by an angle $a <180^o$ clockwise. After that, everything is repeated. Prove that all flags are on the same circle.

2006 Korea Junior Math Olympiad, 3

Tags: geometry , concyclic , perpendicular , parallel

In a circle $O$, there are six points, $A,B,C,D,E, F$ in a counterclockwise order. $BD \perp CF$, and $CF,BE,AD$ are concurrent. Let the perpendicular from $B$ to $AC$ be $M$, and the perpendicular from $D$ to $CE$ be $N$. Prove that $AE // MN$.

Kyiv City MO Seniors Round2 2010+ geometry, 2021.10.4

Tags: concyclic , angle , geometry

Inside the quadrilateral $ABCD$ marked a point $O$ such that $\angle OAD+ \angle OBC = \angle ODA + \angle OCB = 90^o$. Prove that the centers of the circumscribed circles around triangles $OAD$ and $OBC$ as well as the midpoints of the sides $AB$ and $CD$ lie on one circle. (Anton Trygub)

2017 NZMOC Camp Selection Problems, 6

Tags: concyclic , geometry

Let $ABCD$ be a quadrilateral. The circumcircle of the triangle $ABC$ intersects the sides $CD$ and $DA$ in the points $P$ and $Q$ respectively, while the circumcircle of $CDA$ intersects the sides $AB$ and $BC$ in the points $R$ and $S$. The lines $BP$ and $BQ$ intersect the line $RS$ in the points $M$ and $N$ respectively. Prove that the points $M, N, P$ and $Q$ lie on the same circle.

2023 SG Originals, Q3

Tags: geometry , concyclic , concurrency

Let $\vartriangle ABC$ be a triangle with orthocenter $H$, and let $M$, $N$ be the midpoints of $BC$ and $AH$ respectively. Suppose $Q$ is a point on $(ABC)$ such that $\angle AQH = 90^o$. Show that $MN$, the circumcircle of $QNH$, and the $A$-symmedian concur. Note: the $A$-symmedian is the reflection of line $AM$ in the bisector of angle $\angle BAC$.

2011 Brazil Team Selection Test, 2

Tags: geometry , equal segments , concyclic

Given two circles $\omega_1$ and $\omega_2$, with centers $O_1$ and $O_2$, respectively intesrecting at two points $A$ and $B$. Let $X$ and $Y$ be points on $\omega_1$. The lines $XA$ and $YA$ intersect $\omega_2$ again in $Z$ and $W$, respectively, such that $A$ is between $X,Z$ and $A$ is between $Y,W$. Let $M$ be the midpoint of $O_1O_2$, S be the midpoint of $XA$ and $T$ be the midpoint of $WA$. Prove that $MS = MT$ if, and only if, the points $X, Y, Z$ and $W$ are concyclic.

2018 JBMO TST-Turkey, 6

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

A point $E$ is located inside a parallelogram $ABCD$ such that $\angle BAE = \angle BCE$. The centers of the circumcircles of the triangles $ABE,ECB, CDE$ and $DAE$ are concyclic.

Indonesia MO Shortlist - geometry, g8

Tags: geometry , concyclic

Given a circle centered at point $O$, with $AB$ as the diameter. Point $C$ lies on the extension of line $AB$ so that $B$ lies between $A$ and $C$, and the line through $C$ intersects the circle at points $D$ and $E$ (where $D$ lies between $C$ and $E$). $OF$ is the diameter of the circumcircle of triangle $OBD$, and the extension of the line $CF$ intersects the circumcircle of triangle $OBD$ at point $G$. Prove that the points $O, A, E, G$ lie on a circle.

2022-IMOC, G5

Tags: midpoint , concyclic , geometry

$P$ is a point inside $ABC$. $BP$, $CP$ intersect $AC, AB$ at $E, F$, respectively. $AP$ intersect $\odot (ABC)$ again at X. $\odot (ABC)$ and $\odot (AEF)$ intersect again at $S$. $T$ is a point on $BC$ such that $P T \parallel EF$. Prove that $\odot (ST X)$ passes through the midpoint of $BC$. [i]proposed by chengbilly[/i]

Swiss NMO - geometry, 2014.8

Tags: altitude , concyclic , geometry

In the acute-angled triangle $ABC$, let $M$ be the midpoint of the atlitude $h_b$ through $B$ and $N$ be the midpoint of the height $h_c$ through $C$. Further let $P$ be the intersection of $AM$ and $h_c$ and $Q$ be the intersection of $AN$ and $h_b$. Show that $M, N, P$ and $Q$ lie on a circle.

1999 Czech and Slovak Match, 2

Tags: geometry , concyclic

The altitudes through the vertices $A,B,C$ of an acute-angled triangle $ABC$ meet the opposite sides at $D,E,F,$ respectively. The line through $D$ parallel to $EF$ meets the lines $AC$ and $AB$ at $Q$ and $R$, respectively. The line $EF$ meets $BC$ at $P$. Prove that the circumcircle of the triangle $PQR$ passes through the midpoint of $BC$.

2019 Philippine TST, 4

Tags: reflection , concyclic , parallelogram , geometry

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.

2003 Oral Moscow Geometry Olympiad, 3

Tags: equal segments , concyclic , geometry , equal angles

Inside the segment $AC$, an arbitrary point $B$ is selected and circles with diameters $AB$ and $BC$ are constructed. Points $M$ and $L$ are chosen on the circles (in one half-plane with respect to $AC$), respectively, so that $\angle MBA = \angle LBC$. Points $K$ and $F$ are marked, respectively, on rays $BM$ and $BL$ so that $BK = BC$ and $BF = AB$. Prove that points $M, K, F$ and $L$ lie on the same circle.

2013 Tournament of Towns, 3

Tags: equilateral , concyclic , circumcircle , geometry

Let $ABC$ be an equilateral triangle with centre $O$. A line through $C$ meets the circumcircle of triangle $AOB$ at points $D$ and $E$. Prove that points $A, O$ and the midpoints of segments $BD, BE$ are concyclic.

1999 Tournament Of Towns, 2

Tags: concyclic , cyclic , circumcenter , circumcircle , geometry

Let all vertices of a convex quadrilateral $ABCD$ lie on the circumference of a circle with center $O$. Let $F$ be the second intersection point of the circumcircles of the triangles $ABO$ and $CDO$. Prove that the circle passing through the points $A, F$ and $D$ also passes through the intersection point of the segments $AC$ and $BD$. (A Zaslavskiy)

2020 Regional Competition For Advanced Students, 3

Tags: incenter , concyclic , geometry

Let a triangle $ABC$ be given with $AB <AC$. Let the inscribed center of the triangle be $I$. The perpendicular bisector of side $BC$ intersects the angle bisector of $BAC$ at point $S$ and the angle bisector of $CBA$ at point $T$. Prove that the points $C, I, S$ and $T$ lie on a circle. (Karl Czakler)