Found problems: 3882
2004 Tuymaada Olympiad, 3
An acute triangle $ABC$ is inscribed in a circle of radius 1 with centre $O;$ all the angles of $ABC$ are greater than $45^\circ.$
$B_{1}$ is the foot of perpendicular from $B$ to $CO,$ $B_{2}$ is the foot of perpendicular from $B_{1}$ to $AC.$
Similarly, $C_{1}$ is the foot of perpendicular from $C$ to $BO,$ $C_{2}$ is the foot of perpendicular from $C_{1}$ to $AB.$
The lines $B_{1}B_{2}$ and $C_{1}C_{2}$ intersect at $A_{3}.$ The points $B_{3}$ and $C_{3}$ are defined in the same way.
Find the circumradius of triangle $A_{3}B_{3}C_{3}.$
[i]Proposed by F.Bakharev, F.Petrov[/i]
2009 China Team Selection Test, 2
In acute triangle $ ABC,$ points $ P,Q$ lie on its sidelines $ AB,AC,$ respectively. The circumcircle of triangle $ ABC$ intersects of triangle $ APQ$ at $ X$ (different from $ A$). Let $ Y$ be the reflection of $ X$ in line $ PQ.$ Given $ PX>PB.$ Prove that $ S_{\bigtriangleup XPQ}>S_{\bigtriangleup YBC}.$ Where $ S_{\bigtriangleup XYZ}$ denotes the area of triangle $ XYZ.$
2006 Romania Team Selection Test, 1
Let $ABC$ and $AMN$ be two similar triangles with the same orientation, such that $AB=AC$, $AM=AN$ and having disjoint interiors. Let $O$ be the circumcenter of the triangle $MAB$. Prove that the points $O$, $C$, $N$, $A$ lie on the same circle if and only if the triangle $ABC$ is equilateral.
[i]Valentin Vornicu[/i]
2024 Indonesia MO, 3
The triangle $ABC$ has $O$ as its circumcenter, and $H$ as its orthocenter. The line $AH$ and $BH$ intersect the circumcircle of $ABC$ for the second time at points $D$ and $E$, respectively. Let $A'$ and $B'$ be the circumcenters of triangle $AHE$ and $BHD$ respectively. If $A', B', O, H$ are [b]not[/b] collinear, prove that $OH$ intersects the midpoint of segment $A'B'$.
2015 Tournament of Towns, 2
A point $X$ is marked on the base $BC$ of an isosceles $\triangle ABC$, and points $P$ and $Q$ are marked on the sides $AB$ and $AC$ so that $APXQ$ is a parallelogram. Prove that the point $Y$ symmetrical to $X$ with respect to line $PQ$ lies on the circumcircle of the $\triangle ABC$.
[i]($5$ points)[/i]
2005 Polish MO Finals, 2
The points $A, B, C, D$ lie in this order on a circle $o$. The point $S$ lies inside $o$ and has properties $\angle SAD=\angle SCB$ and $\angle SDA= \angle SBC$. Line which in which angle bisector of $\angle ASB$ in included cut the circle in points $P$ and $Q$. Prove $PS =QS$.
2017 South Africa National Olympiad, 5
Let $ABC$ be a triangle with circumcircle $\Gamma$. Let $D$ be a point on segment $BC$ such that $\angle BAD = \angle DAC$, and let $M$ and $N$ be points on segments $BD$ and $CD$, respectively, such that $\angle MAD = \angle DAN$. Let $S, P$ and $Q$ (all different from $A$) be the intersections of the rays $AD$, $AM$ and $AN$ with $\Gamma$, respectively.
Show that the intersection of $SM$ and $QD$ lies on $\Gamma$.
2020 Iran Team Selection Test, 2
Let $O$ be the circumcenter of the triangle $ABC$. Points $D,E$ are on sides $AC,AB$ and points $P,Q,R,S$ are given in plane such that $P,C$ and $R,C$ are on different sides of $AB$ and pints $Q,B$ and $S,B$ are on different sides of $AC$ such that $R,S$ lie on circumcircle of $DAP,EAQ$ and $\triangle BCE \sim \triangle ADQ , \triangle CBD \sim \triangle AEP$(In that order), $\angle ARE=\angle ASD=\angle BAC$, If $RS\| PQ$ prove that $RE ,DS$ are concurrent on $AO$.
[i]Proposed by Alireza Dadgarnia[/i]
2018 Korea Winter Program Practice Test, 1
Let $\Delta ABC$ be a triangle with circumcenter $O$ and circumcircle $w$. Let $S$ be the center of the circle which is tangent with $AB$, $AC$, and $w$ (in the inside), and let the circle meet $w$ at point $K$. Let the circle with diameter $AS$ meet $w$ at $T$. If $M$ is the midpoint of $BC$, show that $K,T,M,O$ are concyclic.
2006 IMAR Test, 3
Consider the isosceles triangle $ABC$ with $AB = AC$, and $M$ the midpoint of $BC$. Find the locus of the points $P$ interior to the triangle, for which $\angle BPM+\angle CPA = \pi$.
2018 Romania Team Selection Tests, 2
Let $ABC$ be a triangle, let $I$ be its incenter, let $\Omega$ be its circumcircle, and let $\omega$ be the $A$- mixtilinear incircle. Let $D,E$ and $T$ be the intersections of $\omega$ and $AB,AC$ and $\Omega$, respectively, let the line $IT$ cross $\omega$ again at $P$, and let lines $PD$ and $PE$ cross the line $BC$ at $M$ and $N$ respectively. Prove that points $D,E,M,N$ are concyclic. What is the center of this circle?
2006 All-Russian Olympiad, 6
Let $K$ and $L$ be two points on the arcs $AB$ and $BC$ of the circumcircle of a triangle $ABC$, respectively, such that $KL\parallel AC$. Show that the incenters of triangles $ABK$ and $CBL$ are equidistant from the midpoint of the arc $ABC$ of the circumcircle of triangle $ABC$.
2023 Korea National Olympiad, 4
Pentagon $ABCDE$ is inscribed in circle $\Omega$. Line $AD$ meets $CE$ at $F$, and $P (\neq E, F)$ is a point on segment $EF$. The circumcircle of triangle $AFP$ meets $\Omega$ at $Q(\neq A)$ and $AC$ at $R(\neq A)$. Line $AD$ meets $BQ$ at $S$, and the circumcircle of triangle $DES$ meets line $BQ, BD$ at $T(\neq S), U(\neq D)$, respectively. Prove that if $F, P, T, S$ are concyclic, then $P, T, R, U$ are concyclic.
2006 MOP Homework, 3
In triangle $ ABC$,$ \angle BAC \equal{} 120^o$. Let the angle bisectors of angles
$ A;B$and $ C$ meet the opposite sides at $ D;E$ and$ F$ respectively.
Prove that the circle on diameter $ EF$ passes through $ D.$
2017 India IMO Training Camp, 1
In an acute triangle $ABC$, points $D$ and $E$ lie on side $BC$ with $BD<BE$. Let $O_1, O_2, O_3, O_4, O_5, O_6$ be the circumcenters of triangles $ABD, ADE, AEC, ABE, ADC, ABC$, respectively. Prove that $O_1, O_3, O_4, O_5$ are con-cyclic if and only if $A, O_2, O_6$ are collinear.
2019 Junior Balkan Team Selection Tests - Romania, 3
Let $d$ be the tangent at $B$ to the circumcircle of the acute scalene triangle $ABC$. Let $K$ be the orthogonal projection of the orthocenter, $H$, of triangle $ABC$ to the line $d$ and $L$ the midpoint of the side $AC$. Prove that the triangle $BKL$ is isosceles.
2008 China Team Selection Test, 1
Let $ ABC$ be a triangle, line $ l$ cuts its sides $ BC,CA,AB$ at $ D,E,F$, respectively. Denote by $ O_{1},O_{2},O_{3}$ the circumcenters of triangle $ AEF,BFD,CDE$, respectively. Prove that the orthocenter of triangle $ O_{1}O_{2}O_{3}$ lies on line $ l$.
2006 Vietnam National Olympiad, 2
Let $ABCD$ be a convex quadrilateral. Take an arbitrary point $M$ on the line $AB$, and let $N$ be the point of intersection of the circumcircles of triangles $MAC$ and $MBC$ (different from $M$). Prove that:
a) The point $N$ lies on a fixed circle;
b) The line $MN$ passes though a fixed point.
1998 Finnish National High School Mathematics Competition, 4
There are $110$ points in a unit square. Show that some four of these points reside in a circle whose radius is $1/8.$
2010 Laurențiu Panaitopol, Tulcea, 3
Let $ R $ be the circumradius of a triangle $ ABC. $ The points $ B,C, $ lie on a circle of radius $ \rho $ that intersects $ AB,AC $ at $ E,D, $ respectively. $ \rho' $ is the circumradius of $ ADE. $ Show that there exists a triangle with sides $ R,\rho ,\rho' , $ and having an angle whose value doesn't depend on $ \rho . $
[i]Laurențiu Panaitopol[/i]
2011 Greece Team Selection Test, 4
Let $ABCD$ be a cyclic quadrilateral and let $K,L,M,N,S,T$ the midpoints of $AB, BC, CD, AD, AC, BD$ respectively. Prove that the circumcenters of $KLS, LMT, MNS, NKT$ form a cyclic quadrilateral which is similar to $ABCD$.
2014 Ukraine Team Selection Test, 4
The $A$-excircle of the triangle $ABC$ touches the side $BC$ at point $K$. The circumcircles of triangles $AKB$ and $AKC$ intersect for the second time with the bisector of angle $A$ at points $X$ and $Y$ respectively. Let $M$ be the midpoint of $BC$. Prove that the circumcenter of triangle $XYM$ lies on $BC$.
2001 China National Olympiad, 1
Let $a$ be real number with $\sqrt{2}<a<2$, and let $ABCD$ be a convex cyclic quadrilateral whose circumcentre $O$ lies in its interior. The quadrilateral's circumcircle $\omega$ has radius $1$, and the longest and shortest sides of the quadrilateral have length $a$ and $\sqrt{4-a^2}$, respectively. Lines $L_A,L_B,L_C,L_D$ are tangent to $\omega$ at $A,B,C,D$, respectively.
Let lines $L_A$ and $L_B$, $L_B$ and $L_C$,$L_C$ and $L_D$,$L_D$ and $L_A$ intersect at $A',B',C',D'$ respectively. Determine the minimum value of $\frac{S_{A'B'C'D'}}{S_{ABCD}}$.
2017 Saudi Arabia BMO TST, 3
Let $ABC$ be an acute triangle and $(O)$ be its circumcircle. Denote by $H$ its orthocenter and $I$ the midpoint of $BC$. The lines $BH, CH$ intersect $AC,AB$ at $E, F$ respectively. The circles $(IBF$) and $(ICE)$ meet again at $D$.
a) Prove that $D, I,A$ are collinear and $HD, EF, BC$ are concurrent.
b) Let $L$ be the foot of the angle bisector of $\angle BAC$ on the side $BC$. The circle $(ADL)$ intersects $(O)$ again at $K$ and intersects the line $BC$ at $S$ out of the side $BC$. Suppose that $AK,AS$ intersects the circles $(AEF)$ again at $G, T$ respectively. Prove that $TG = TD$.
2018 Korea Winter Program Practice Test, 2
Let $\Delta ABC$ be a triangle and $P$ be a point in its interior. Prove that \[ \frac{[BPC]}{PA^2}+\frac{[CPA]}{PB^2}+\frac{[APB]}{PC^2} \ge \frac{[ABC]}{R^2} \]
where $R$ is the radius of the circumcircle of $\Delta ABC$, and $[XYZ]$ is the area of $\Delta XYZ$.