Found problems: 257
2016 All-Russian Olympiad, 2
$\omega$ is a circle inside angle $\measuredangle BAC$ and it is tangent to sides of this angle at $B,C$.An arbitrary line $ \ell $ intersects with $AB,AC$ at $K,L$,respectively and intersect with $\omega$ at $P,Q$.Points $S,T$ are on $BC$ such that $KS \parallel AC$ and $TL \parallel AB$.Prove that $P,Q,S,T$ are concyclic.(I.Bogdanov,P.Kozhevnikov)
2012 Korea Junior Math Olympiad, 2
A pentagon $ABCDE$ is inscribed in a circle $O$, and satis fies $\angle A = 90^o, AB = CD$. Let $F$ be a point on segment $AE$. Let $BF$ hit $O$ again at $J(\ne B)$, $CE \cap DJ = K$, $BD\cap FK = L$. Prove that $B,L,E,F$ are cyclic.
Russian TST 2022, P2
The quadrilateral $ABCD$ is inscribed in the circle $\Gamma$. Let $I_B$ and $I_D$ be the centers of the circles $\omega_B$ and $\omega_D$ inscribed in the triangles $ABC$ and $ADC$, respectively. A common external tangent to $\omega_B$ and $\omega_D$ intersects $\Gamma$ at $K$ and $L{}$. Prove that $I_B,I_D,K$ and $L{}$ lie on the same circle.
2023 Sharygin Geometry Olympiad, 9.1
The ratio of the median $AM$ of a triangle $ABC$ to the side $BC$ equals $\sqrt{3}:2$. The points on the sides of $ABC$ dividing these side into $3$ equal parts are marked. Prove that some $4$ of these $6$ points are concyclic.
2015 Costa Rica - Final Round, 1
Let $ABCD$ be a quadrilateral whose diagonals are perpendicular, and let $S$ be the intersection of those diagonals. Let $K, L, M$ and $N$ be the reflections of $S$ on the sides $AB$, $BC$, $CD$ and $DA$ respectively. $BN$ cuts the circumcircle of $\vartriangle SKN$ at $E$ and $BM$ cuts the circumcircle of $\vartriangle SLM$ at $F$. Prove that the quadrilateral $EFLK$ is cyclic.
2010 All-Russian Olympiad Regional Round, 10.3
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.
1975 Chisinau City MO, 111
Three squares are constructed on the sides of the triangle to the outside. What should be the angles of the triangle so that the six vertices of these squares, other than the vertices of the triangle, lie on the same circle?
2021 Middle European Mathematical Olympiad, 3
Let $ABC$ be an acute triangle and $D$ an interior point of segment $BC$. Points $E$ and $F$ lie in the half-plane determined by the line $BC$ containing $A$ such that $DE$ is perpendicular to $BE$ and $DE$ is tangent to the circumcircle of $ACD$, while $DF$ is perpendicular to $CF$ and $DF$ is tangent to the circumcircle of $ABD$. Prove that the points $A, D, E$ and $F$ are concyclic.
2013 Sharygin Geometry Olympiad, 4
A point $F$ inside a triangle $ABC$ is chosen so that $\angle AFB = \angle BFC = \angle CFA$. The line passing through $F$ and perpendicular to $BC$ meets the median from $A$ at point $A_1$. Points $B_1$ and $C_1$ are defined similarly. Prove that the points $A_1, B_1$, and $C_1$ are three vertices of some regular hexagon, and that the three remaining vertices of that hexagon lie on the sidelines of $ABC$.
2020 Kosovo National Mathematical Olympiad, 3
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.
2022-IMOC, G3
Let $\vartriangle ABC$ be an acute triangle. $R$ is a point on arc $BC$. Choose two points $P, Q$ on $AR$ such that $B,P,C,Q$ are concyclic. Let the second intersection of $BP$, $CP$, $BQ$, $CQ$ and the circumcircle of $\vartriangle ABC$ is $P_B$, $P_C$, $Q_B$, $Q_C$, respectively. Let the circumcenter of $\vartriangle P P_BP_C$ and $\vartriangle QQ_BQ_C$ are $O_P$ and $O_Q$, respectively. Prove that $A,O_P,O_Q,R$ are concylic.
[i]proposed by andychang[/i]
2011 Brazil Team Selection Test, 2
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.
2022 Oral Moscow Geometry Olympiad, 2
In an acute triangle $ABC$,$O$ is the center of the circumscribed circle $\omega$, $P$ is the point of intersection of the tangents to $\omega$ through the points $B$ and $C$, the median AM intersects the circle $\omega$ at point $D$. Prove that points $A, D, P$ and $O$ lie on the same circle.
(D. Prokopenko)
2018 Regional Olympiad of Mexico Northwest, 3
Let $ABC$ be an acute triangle orthocenter angle $H$. Let $\omega_1$ be the circle tangent to $BC$ at $B$ and passing through $H$ and $\omega_2$ the circle tangent to $BC$ at $C$ and passing through through $H$. A line $\ell$ passing through $H$ intersects the circles $\omega_1$ and $\omega_2$ at points $D$ and $E$, respectively (with $D$ and $E$ other than $H$). Lines $BD$ and $CE$ intersect at $F$, the lines $\ell$ and $AF$ intersect at $X$ and the circles $\omega_1$ and $\omega_2$ intersect at the points $P$ and $H$. Prove that the points $A, H, P$ and $X$ are still on the same circle.
2013 Saudi Arabia IMO TST, 2
Let $ABC$ be an acute triangle, and let $AA_1, BB_1$, and $CC_1$ be its altitudes. Segments $AA_1$ and $B_1C_1$ meet at point $K$. The perpendicular bisector of segment $A_1K$ intersects sides $AB$ and $AC$ at $L$ and $M$, respectively. Prove that points $A,A_1, L$, and $M$ lie on a circle.
2003 Oral Moscow Geometry Olympiad, 3
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.
Indonesia MO Shortlist - geometry, g2
It is known that two circles have centers at $P$ and $Q$. Prove that the intersection points of the two internal common tangents of the two circles with their two external common tangents lie on the same circle.
Geometry Mathley 2011-12, 12.4
Quadrilateral$ ABCD$ has two diagonals $AC,BD$ that are mutually perpendicular. Let $M$ be the Miquel point of the complete quadrilateral formed by lines $AB,BC,CD,DA$. Suppose that $L$ is the intersection of two circles $(MAC)$ and $(MBD)$. Prove that the circumcenters of triangles $LAB,LBC,LCD,LDA$ are on the same circle called $\omega$ and that three circles $(MAC), (MBD), \omega$ are pairwise orthogonal.
Nguyễn Văn Linh
Ukrainian TYM Qualifying - geometry, 2020.12
On the side $CD$ of the square $ABCD$, the point $F$ is chosen and the equal squares $DGFE$ and $AKEH$ are constructed ($E$ and $H$ lie inside the square). Let $M$ be the midpoint of $DF$, $J$ is the incenter of the triangle $CFH$. Prove that:
a) the points $D, K, H, J, F$ lie on the same circle;
b) the circles inscribed in triangles $CFH$ and $GMF$ have the same radii.
2006 Sharygin Geometry Olympiad, 26
Four cones are given with a common vertex and the same generatrix, but with, generally speaking, different radii of the bases. Each of them is tangent to two others. Prove that the four tangent points of the circles of the bases of the cones lie on the same circle.
2022-IMOC, G5
$P$ 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]
1976 All Soviet Union Mathematical Olympiad, 226
Given regular $1976$-gon. The midpoints of all the sides and diagonals are marked. What is the greatest number of the marked points lying on one circumference?
2012 Dutch BxMO/EGMO TST, 2
Let $\triangle ABC$ be a triangle and let $X$ be a point in the interior of the triangle. The second intersection points of the lines $XA,XB$ and $XC$ with the circumcircle of $\triangle ABC$ are $P,Q$ and $R$. Let $U$ be a point on the ray $XP$ (these are the points on the line $XP$ such that $P$ and $U$ lie on the same side of $X$). The line through $U$ parallel to $AB$ intersects $BQ$ in $V$ . The line through $U$ parallel to $AC$ intersects $CR$ in $W$. Prove that $Q, R, V$ , and $W$ lie on a circle.
2021 Sharygin Geometry Olympiad, 9.5
Let $O$ be the clrcumcenter of triangle $ABC$. Points $X$ and $Y$ on side $BC$ are such that $AX = BX$ and $AY = CY$. Prove that the circumcircle of triangle $AXY$ passes through the circumceuters of triangles $AOB$ and $AOC$.
2014 Belarus Team Selection Test, 1
Circles $\Gamma_1$ and $\Gamma_2$ meet at points $X$ and $Y$. A circle $S_1$ touches internally $\Gamma_1$ at $A$ and $\Gamma_2$ externally at $B$. A circle $S_2$ touches $\Gamma_2$ internally at $C$ and $\Gamma_1$ externally at $D$. Prove that the points $A, B, C, D$ are either collinear or concyclic.
(A. Voidelevich)