Found problems: 3882
1999 Bosnia and Herzegovina Team Selection Test, 2
Prove the inequality $$\frac{a^2}{b+c-a}+\frac{b^2}{a+c-b}+\frac{c^2}{a+b-c} \geq 3\sqrt{3}R$$ in triangle $ABC$ where $a$, $b$ and $c$ are sides of triangle and $R$ radius of circumcircle of $ABC$
2008 Indonesia MO, 1
Given triangle $ ABC$. Points $ D,E,F$ outside triangle $ ABC$ are chosen such that triangles $ ABD$, $ BCE$, and $ CAF$ are equilateral triangles. Prove that cicumcircles of these three triangles are concurrent.
2016 Peru IMO TST, 3
Let $ABCD$ a convex quadrilateral such that $AD$ and $BC$ are not parallel. Let $M$ and $N$ the midpoints of $AD$ and $BC$ respectively. The segment $MN$ intersects $AC$ and $BD$ in $K$ and $L$ respectively, Show that at least one point of the intersections of the circumcircles of $AKM$ and $BNL$ is in the line $AB$.
2014 ELMO Shortlist, 3
Let $A_1A_2A_3 \cdots A_{2013}$ be a cyclic $2013$-gon. Prove that for every point $P$ not the circumcenter of the $2013$-gon, there exists a point $Q\neq P$ such that $\frac{A_iP}{A_iQ}$ is constant for $i \in \{1, 2, 3, \cdots, 2013\}$.
[i]Proposed by Robin Park[/i]
2002 IMO, 2
The circle $S$ has centre $O$, and $BC$ is a diameter of $S$. Let $A$ be a point of $S$ such that $\angle AOB<120{{}^\circ}$. Let $D$ be the midpoint of the arc $AB$ which does not contain $C$. The line through $O$ parallel to $DA$ meets the line $AC$ at $I$. The perpendicular bisector of $OA$ meets $S$ at $E$ and at $F$. Prove that $I$ is the incentre of the triangle $CEF.$
2005 Germany Team Selection Test, 2
Let $ABC$ be a triangle satisfying $BC < CA$. Let $P$ be an arbitrary point on the side $AB$ (different from $A$ and $B$), and let the line $CP$ meet the circumcircle of triangle $ABC$ at a point $S$ (apart from the point $C$).
Let the circumcircle of triangle $ASP$ meet the line $CA$ at a point $R$ (apart from $A$), and let the circumcircle of triangle $BPS$ meet the line $CB$ at a point $Q$ (apart from $B$).
Prove that the excircle of triangle $APR$ at the side $AP$ is identical with the excircle of triangle $PQB$ at the side $PQ$ if and only if the point $S$ is the midpoint of the arc $AB$ on the circumcircle of triangle $ABC$.
2002 Austrian-Polish Competition, 6
The diagonals of a convex quadrilateral $ABCD$ intersect in the point $E$. Let $U$ be the circumcenter of the triangle $ABE$ and $H$ be its orthocenter. Similarly, let $V$ be the circumcenter of the triangle $CDE$ and $K$ be its orthocenter. Prove that $E$ lies on the line $UK$ if and only if it lies on the line $VH$.
2012 Balkan MO Shortlist, G3
Let $ABC$ be a triangle with circumcircle $c$ and circumcenter $O$, and let $D$ be a point on the side $BC$ different from the vertices and the midpoint of $BC$. Let $K$ be the point where the circumcircle $c_1$ of the triangle $BOD$ intersects $c$ for the second time and let $Z$ be the point where $c_1$ meets the line $AB$. Let $M$ be the point where the circumcircle $c_2$ of the triangle $COD$ intersects $c$ for the second time and let $E$ be the point where $c_2$ meets the line $AC$. Finally let $N$ be the point where the circumcircle $c_3$ of the triangle $AEZ$ meets $c$ again. Prove that the triangles $ABC$ and $NKM$ are congruent.
2005 Germany Team Selection Test, 2
Let $ABC$ be a triangle satisfying $BC < CA$. Let $P$ be an arbitrary point on the side $AB$ (different from $A$ and $B$), and let the line $CP$ meet the circumcircle of triangle $ABC$ at a point $S$ (apart from the point $C$).
Let the circumcircle of triangle $ASP$ meet the line $CA$ at a point $R$ (apart from $A$), and let the circumcircle of triangle $BPS$ meet the line $CB$ at a point $Q$ (apart from $B$).
Prove that the excircle of triangle $APR$ at the side $AP$ is identical with the excircle of triangle $PQB$ at the side $PQ$ if and only if the point $S$ is the midpoint of the arc $AB$ on the circumcircle of triangle $ABC$.
2012 Turkey Team Selection Test, 1
In a triangle $ABC,$ incircle touches the sides $BC, CA, AB$ at $D, E, F,$ respectively. A circle $\omega$ passing through $A$ and tangent to line $BC$ at $D$ intersects the line segments $BF$ and $CE$ at $K$ and $L,$ respectively. The line passing through $E$ and parallel to $DL$ intersects the line passing through $F$ and parallel to $DK$ at $P.$ If $R_1, R_2, R_3, R_4$ denotes the circumradius of the triangles $AFD, AED, FPD, EPD,$ respectively, prove that $R_1R_4=R_2R_3.$
2020 South East Mathematical Olympiad, 2
In a scalene triangle $\Delta ABC$, $AB<AC$, $PB$ and $PC$ are tangents of the circumcircle $(O)$ of $\Delta ABC$. A point $R$ lies on the arc $\widehat{AC}$(not containing $B$), $PR$ intersects $(O)$ again at $Q$. Suppose $I$ is the incenter of $\Delta ABC$, $ID \perp BC$ at $D$, $QD$ intersects $(O)$ again at $G$. A line passing through $I$ and perpendicular to $AI$ intersects $AB,AC$ at $M,N$, respectively.
Prove that, if $AR \parallel BC$, then $A,G,M,N$ are concyclic.
2003 China Team Selection Test, 1
In triangle $ABC$, $AB > BC > CA$, $AB=6$, $\angle{B}-\angle{C}=90^o$. The incircle touches $BC$ at $E$ and $EF$ is a diameter of the incircle. Radical $AF$ intersect $BC$ at $D$. $DE$ equals to the circumradius of $\triangle{ABC}$. Find $BC$ and $AC$.
Ukraine Correspondence MO - geometry, 2005.11
Let the circle $\omega$ be circumscribed around the triangle $\vartriangle ABC$ with right angle $\angle A$. Tangent to the circle $\omega$ at point $A$ intersects the line $BC$ at point $D$. Point $E$ is symmetric to $A$ with respect to the line $BC$. Let $K$ be the foot of the perpendicular drawn from point $A$ on $BE$, $L$ the midpoint of $AK$. The line $BL$ intersects the circle $\omega$ for the second time at the point $N$. Prove that the line $BD$ is tangent to the circle circumscribed around the triangle $\vartriangle ADM$.
1988 USAMO, 4
Let $I$ be the incenter of triangle $ABC$, and let $A'$, $B'$, and $C'$ be the circumcenters of triangles $IBC$, $ICA$, and $IAB$, respectively. Prove that the circumcircles of triangles $ABC$ and $A'B'C'$ are concentric.
2002 All-Russian Olympiad Regional Round, 11.7
Given a convex quadrilateral $ABCD$.Let $\ell_A,\ell_B,\ell_C,\ell_D$ be exterior angle bisectors of quadrilateral $ABCD$.
Let $\ell_A \cap \ell_B=K,\ell_B \cap \ell_C=L,\ell_C \cap \ell_D=M,\ell_D \cap \ell_A=N$.Prove that if circumcircles of triangles $ABK$ and $CDM$ be externally tangent to each other then circumcircles of the triangles $BCL$ and $DAN$ are externally tangent to each other.(L.Emelyanov)
Kharkiv City MO Seniors - geometry, 2018.11.4
The line $\ell$ parallel to the side $BC$ of the triangle $ABC$, intersects its sides $AB,AC$ at the points $D,E$, respectively. The circumscribed circle of triangle $ABC$ intersects line $\ell$ at points $F$ and $G$, such that points $F,D,E,G$ lie on line $\ell$ in this order. The circumscribed circles of the triangles $FEB$ and $DGC$ intersect at points $P$ and $Q$. Prove that points $A, P$ and $Q$ are collinear.
2016 Oral Moscow Geometry Olympiad, 6
Given an acute triangle $ABC$. Let $A'$ be a point symmetric to $A$ with respect to $BC, O_A$ is the center of the circle passing through $A$ and the midpoints of the segments $A'B$ and $A'C. O_B$ and $O_C$ points are defined similarly. Find the ratio of the radii of the circles circumscribed around the triangles $ABC$ and $O_AO_BO_C$.
2023 Vietnam National Olympiad, 7
Let $\triangle{ABC}$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Incircle $(I)$ of the $\triangle{ABC}$ is tangent to the sides $BC,CA,AB$ at $M,N,P$ respectively. Denote $\Omega_A$ to be the circle passing through point $A$, external tangent to $(I)$ at $A'$ and cut again $AB,AC$ at $A_b,A_c$ respectively. The circles $\Omega_B,\Omega_C$ and points $B',B_a,B_c,C',C_a,C_b$ are defined similarly.
$a)$ Prove $B_cC_b+C_aA_c+A_bB_a \ge NP+PM+MN$.
$b)$ Suppose $A',B',C'$ lie on $AM,BN,CP$ respectively. Denote $K$ as the circumcenter of the triangle formed by lines $A_bA_c,B_cB_a,C_aC_b.$ Prove $OH//IK$.
2024 Korea Junior Math Olympiad (First Round), 20.
There is a $\triangle ABC$ which $\angle C=90$, and $\bar{AB}=36$
On the circumcircle of $\triangle ABC$, there is $\overarc{BC}$ which does not include point $A$.
D is on $\overarc{BC}$. It satisfies $2\times\angle CAD = \angle BAD $
$E: \bar{AD}\cap\bar{BC} $ $ \bar{AE}=20 $
Find $ \bar{BD}^2 $
2019 Latvia Baltic Way TST, 11
Let $A_1A_2...A_{2018}$ be regular $2018$-gon. Radius of it's circumcircle is $R$. Prove that:
$$A_1A_{1008}-A_1A_{1006}+A_1A_{1004}-A_1A_{1002} + ... + A_1A_4 -A_1A_2=R$$
2021 Middle European Mathematical Olympiad, 5
Let $AD$ be the diameter of the circumcircle of an acute triangle $ABC$. The lines through $D$ parallel to $AB$ and $AC$ meet lines $AC$ and $AB$ in points $E$ and $F$, respectively. Lines $EF$ and $BC$ meet at $G$. Prove that $AD$ and $DG$ are perpendicular.
2006 Costa Rica - Final Round, 3
Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle.
[i]Proposed by Dimitris Kontogiannis, Greece[/i]
2010 Iran Team Selection Test, 11
Let $O, H$ be circumcenter and orthogonal center of triangle $ABC$. $M,N$ are midpoints of $BH$ and $CH$. $BB'$ is diagonal of circumcircle. If $HONM$ is a cyclic quadrilateral, prove that $B'N=\frac12AC$.
2002 China Team Selection Test, 2
Circles $ \omega_{1}$ and $ \omega_{2}$ intersect at points $ A$ and $ B.$ Points $ C$ and $ D$ are on circles $ \omega_{1}$ and $ \omega_{2},$ respectively,
such that lines $ AC$ and $ AD$ are tangent to circles $ \omega_{2}$ and $ \omega_{1},$ respectively.
Let $ I_{1}$ and $ I_{2}$ be the incenters of triangles $ ABC$ and $ ABD,$ respectively. Segments $ I_{1}I_{2}$ and $ AB$ intersect at $ E$.
Prove that: $ \frac {1}{AE} \equal{} \frac {1}{AC} \plus{} \frac {1}{AD}$
Kyiv City MO Seniors 2003+ geometry, 2009.10.4
In the triangle $ABC$ the angle bisectors $AL$ and $BT$ are drawn, which intersect at the point $I$, and their extensions intersect the circle circumscribed around the triangle $ABC$ at the points $E$ and $D$ respectively. The segment $DE$ intersects the sides $AC$ and $BC$ at the points $F$ and $K$, respectively. Prove that:
a) quadrilateral $IKCF$ is rhombus;
b) the side of this rhombus is $\sqrt {DF \cdot EK}$.
(Rozhkova Maria)