Found problems: 3882
2000 Bosnia and Herzegovina Team Selection Test, 2
Let $S$ be a point inside triangle $ABC$ and let lines $AS$, $BS$ and $CS$ intersect sides $BC$, $CA$ and $AB$ in points $X$, $Y$ and $Z$, respectively. Prove that $$\frac{BX\cdot CX}{AX^2}+\frac{CY\cdot AY}{BY^2}+\frac{AZ\cdot BZ}{CZ^2}=\frac{R}{r}-1$$ iff $S$ is incenter of $ABC$
2013 Saudi Arabia GMO TST, 3
$ABC$ is a triangle, $H$ its orthocenter, $I$ its incenter, $O$ its circumcenter and $\omega$ its circumcircle. Line $CI$ intersects circle $\omega$ at point $D$ different from $C$. Assume that $AB = ID$ and $AH = OH$. Find the angles of triangle $ABC$.
2004 Iran MO (3rd Round), 9
Let $ABC$ be a triangle, and $O$ the center of its circumcircle.
Let a line through the point $O$ intersect the lines $AB$ and $AC$ at the points $M$ and $N$, respectively. Denote by $S$ and $R$ the midpoints of the segments $BN$ and $CM$, respectively.
Prove that $\measuredangle ROS=\measuredangle BAC$.
2004 Polish MO Finals, 1
A point $ D$ is taken on the side $ AB$ of a triangle $ ABC$. Two circles passing through $ D$ and touching $ AC$ and $ BC$ at $ A$ and $ B$ respectively intersect again at point $ E$. Let $ F$ be the point symmetric to $ C$ with respect to the perpendicular bisector of $ AB$. Prove that the points $ D,E,F$ lie on a line.
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)
1990 IMO Longlists, 7
Let $S$ be the incenter of triangle $ABC$. $A_1, B_1, C_1$ are the intersections of $AS, BS, CS$ with the circumcircle of triangle $ABC$ respectively. Prove that $SA_1 + SB_1 + SC_1 \geq SA + SB + SC.$
1989 Bulgaria National Olympiad, Problem 5
Prove that the perpendiculars, drawn from the midpoints of the edges of the base of a given tetrahedron to the opposite lateral edges, have a common point if and only if the circumcenter of the tetrahedron, the centroid of the base, and the top vertex of the tetrahedron are collinear.
Kyiv City MO Seniors Round2 2010+ geometry, 2010.10.4
The points $A \ne B$ are given on the plane. The point $C$ moves along the plane in such a way that $\angle ACB = \alpha$ , where $\alpha$ is the fixed angle from the interval ($0^o, 180^o$). The circle inscribed in triangle $ABC$ has center the point $I$ and touches the sides $AB, BC, CA$ at points $D, E, F$ accordingly. Rays $AI$ and $BI$ intersect the line $EF$ at points $M$ and $N$, respectively. Show that:
a) the segment $MN$ has a constant length,
b) all circles circumscribed around triangle $DMN$ have a common point
2010 ELMO Shortlist, 5
Determine all (not necessarily finite) sets $S$ of points in the plane such that given any four distinct points in $S$, there is a circle passing through all four or a line passing through some three.
[i]Carl Lian.[/i]
2003 Tournament Of Towns, 3
Point $M$ is chosen in triangle $ABC$ so that the radii of the circumcircles of triangles $AMC, BMC$, and $BMA$ are no smaller than the radius of the circumcircle of $ABC$. Prove that all four radii are equal.
2014 Dutch IMO TST, 3
Let $H$ be the orthocentre of an acute triangle $ABC$. The line through $A$ perpendicular to $AC$ and the line through $B$ perpendicular to $BC$ intersect in $D$. The circle with centre $C$ through $H$ intersects the circumcircle of triangle $ABC$ in the points $E$ and $F$. Prove that $|DE| = |DF| = |AB|$.
2018 China Team Selection Test, 5
Let $ABC$ be a triangle with $\angle BAC > 90 ^{\circ}$, and let $O$ be its circumcenter and $\omega$ be its circumcircle. The tangent line of $\omega$ at $A$ intersects the tangent line of $\omega$ at $B$ and $C$ respectively at point $P$ and $Q$. Let $D,E$ be the feet of the altitudes from $P,Q$ onto $BC$, respectively. $F,G$ are two points on $\overline{PQ}$ different from $A$, so that $A,F,B,E$ and $A,G,C,D$ are both concyclic. Let M be the midpoint of $\overline{DE}$. Prove that $DF,OM,EG$ are concurrent.
2024 Middle European Mathematical Olympiad, 5
Let $ABC$ be a triangle with $\angle BAC=60^\circ$. Let $D$ be a point on the line $AC$ such that $AB = AD$ and $A$ lies between $C$ and $D$. Suppose that there are two points $E \ne F$ on the circumcircle of the triangle $DBC$ such that $AE = AF = BC$. Prove that the line $EF$ passes through the circumcenter of $ABC$.
2009 Iran MO (3rd Round), 3
3-There is given a trapezoid $ ABCD$ in the plane with $ BC\parallel{}AD$.We know that the angle bisectors of the angles of the trapezoid are concurrent at $ O$.Let $ T$ be the intersection of the diagonals $ AC,BD$.Let $ Q$ be on $ CD$ such that $ \angle OQD \equal{} 90^\circ$.Prove that if the circumcircle of the triangle $ OTQ$ intersects $ CD$ again at $ P$ then $ TP\parallel{}AD$.
2005 Korea Junior Math Olympiad, 2
For triangle $ABC, P$ and $Q$ satisfy $\angle BPA + \angle AQC = 90^o$. It is provided that the vertices of the triangle $BAP$ and $ACQ$ are ordered counterclockwise (or clockwise). Let the intersection of the circumcircles of the two triangles be $N$ ($A \ne N$, however if $A$ is the only intersection $A = N$), and the midpoint of segment $BC$ be $M$. Show that the length of $MN$ does not depend on $P$ and $Q$.
2009 Indonesia TST, 3
Let $ ABC$ be an isoceles triangle with $ AC\equal{}BC$. A point $ P$ lies inside $ ABC$ such that \[ \angle PAB \equal{} \angle PBC, \angle PAC \equal{} \angle PCB.\] Let $ M$ be the midpoint of $ AB$ and $ K$ be the intersection of $ BP$ and $ AC$. Prove that $ AP$ and $ PK$ trisect $ \angle MPC$.
1994 Baltic Way, 16
The Wonder Island is inhabited by Hedgehogs. Each Hedgehog consists of three segments of unit length having a common endpoint, with all three angles between them $120^{\circ}$. Given that all Hedgehogs are lying flat on the island and no two of them touch each other, prove that there is a finite number of Hedgehogs on Wonder Island.
2017 Costa Rica - Final Round, G4
In triangle $ABC$ with incenter $I$ and circumcircle $\omega$, the tangent through $C$ to $\omega$ intersects $AB$ at point $D$. The angle bisector of $\angle CDB$ intersects $AI$ and $BI$ at $E$ and $F$, respectively. Let $M$ be the midpoint of $[EF]$. Prove that line $MI$ passes through the midpoint of arc $ACB$ of $w$ .
2001 Baltic Way, 10
In a triangle $ABC$, the bisector of $\angle BAC$ meets the side $BC$ at the point $D$. Knowing that $|BD|\cdot |CD|=|AD|^2$ and $\angle ADB=45^{\circ}$, determine the angles of triangle $ABC$.
2018 PUMaC Live Round, 8.2
The triangle $ABC$ satisfies $AB=10$ and has angles $\angle{A}=75^{\circ}$, $\angle{B}=60^{\circ}$, and $\angle C = 45^{\circ}$. Let $I_A$ be the center of the excircle opposite $A$, and let $D$, $E$ be the circumcenters of triangle $BCI_A$ and $ACI_A$ respectively. If $O$ is the circumcenter of triangle $ABC$, then the area of triangle $EOD$ can be written as $\tfrac{a\sqrt{b}}{c}$ for square-free $b$ and coprime $a,c$. Find the value of $a+b+c$.
2019 Iran Team Selection Test, 4
Given an acute-angled triangle $ABC$ with orthocenter $H$. Reflection of nine-point circle about $AH$ intersects circumcircle at points $X$ and $Y$. Prove that $AH$ is the external bisector of $\angle XHY$.
[i]Proposed by Mohammad Javad Shabani[/i]
2019 Azerbaijan BMO TST, 2
Let $ABC$ be a triangle inscribed in circle $\Gamma$ with center $O$. Let $H$ be the orthocenter of triangle $ABC$ and let $K$ be the midpoint of $OH$. Tangent of $\Gamma$ at $B$ intersects the perpendicular bisector of $AC$ at $L$. Tangent of $\Gamma$ at $C$ intersects the perpendicular bisector of $AB$ at $M$. Prove that $AK$ and $LM$ are perpendicular.
by Michael Sarantis, Greece
2017 USA Team Selection Test, 2
Let $ABC$ be a triangle with altitude $\overline{AE}$. The $A$-excircle touches $\overline{BC}$ at $D$, and intersects the circumcircle at two points $F$ and $G$. Prove that one can select points $V$ and $N$ on lines $DG$ and $DF$ such that quadrilateral $EVAN$ is a rhombus.
[i]Danielle Wang and Evan Chen[/i]
2005 Taiwan TST Round 2, 2
Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$.
[i]Proposed by Hojoo Lee, Korea[/i]
2013 Dutch IMO TST, 2
Let $P$ be the point of intersection of the diagonals of a convex quadrilateral $ABCD$.Let $X,Y,Z$ be points on the interior of $AB,BC,CD$ respectively such that $\frac{AX}{XB}=\frac{BY}{YC}=\frac{CZ}{ZD}=2$. Suppose that $XY$ is tangent to the circumcircle of $\triangle CYZ$ and that $Y Z$ is tangent to the circumcircle of $\triangle BXY$.Show that $\angle APD=\angle XYZ$.