Found problems: 230
May Olympiad L2 - geometry, 2009.2
Let $ABCD$ be a convex quadrilateral such that the triangle $ABD$ is equilateral and the triangle $BCD$ is isosceles, with $\angle C = 90^o$. If $E$ is the midpoint of the side $AD$, determine the measure of the angle $\angle CED$.
2013 Sharygin Geometry Olympiad, 21
Chords $BC$ and $DE$ of circle $\omega$ meet at point $A$. The line through $D$ parallel to $BC$ meets $\omega$ again at $F$, and $FA$ meets $\omega$ again at $T$. Let $M = ET \cap BC$ and let $N$ be the reflection of $A$ over $M$. Show that $(DEN)$ passes through the midpoint of $BC$.
2011 Dutch IMO TST, 3
The circles $\Gamma_1$ and $\Gamma_2$ intersect at $D$ and $P$. The common tangent line of the two circles closest to point $D$ touches $\Gamma_1$ in A and $\Gamma_2$ in $B$. The line $AD$ intersects $\Gamma_2$ for the second time in $C$. Let $M$ be the midpoint of line segment $BC$. Prove that $\angle DPM = \angle BDC$.
Brazil L2 Finals (OBM) - geometry, 2005.6
The angle $B$ of a triangle $ABC$ is $120^o$. Let $M$ be a point on the side $AC$ and $K$ a point on the extension of the side $AB$, such that $BM$ is the internal bisector of the angle $\angle ABC$ and $CK$ is the external bisector corresponding to the angle $\angle ACB$ . The segment $MK$ intersects $BC$ at point $P$. Prove that $\angle APM = 30^o$.
2013 Purple Comet Problems, 26
The diagram below shows the first three figures of a sequence of figures. The fi rst figure shows an equilateral triangle $ABC$ with side length $1$. The leading edge of the triangle going in a clockwise direction around $A$ is labeled $\overline{AB}$ and is darkened in on the figure. The second figure shows the same equilateral triangle with a square with side length $1$ attached to the leading clockwise edge of the triangle. The third figure shows the same triangle and square with a regular pentagon with side length $1$ attached to the leading clockwise edge of the square. The fourth fi gure in the sequence will be formed by attaching a regular hexagon with side length $1$ to the leading clockwise edge of the pentagon. The hexagon will overlap the triangle. Continue this sequence through the eighth figure. After attaching the last regular figure (a regular decagon), its leading clockwise edge will form an angle of less than $180^\circ$ with the side $\overline{AC}$ of the equilateral triangle. The degree measure of that angle can be written in the form $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[asy]
size(250);
defaultpen(linewidth(0.7)+fontsize(10));
pair x[],y[],z[];
x[0]=origin;
x[1]=(5,0);
x[2]=rotate(60,x[0])*x[1];
draw(x[0]--x[1]--x[2]--cycle);
for(int i=0;i<=2;i=i+1)
{
y[i]=x[i]+(15,0);
}
y[3]=rotate(90,y[0])*y[2];
y[4]=rotate(-90,y[2])*y[0];
draw(y[0]--y[1]--y[2]--y[0]--y[3]--y[4]--y[2]);
for(int i=0;i<=4;i=i+1)
{
z[i]=y[i]+(15,0);
}
z[5]=rotate(108,z[4])*z[2];
z[6]=rotate(108,z[5])*z[4];
z[7]=rotate(108,z[6])*z[5];
draw(z[0]--z[1]--z[2]--z[0]--z[3]--z[4]--z[2]--z[7]--z[6]--z[5]--z[4]);
dot(x[2]^^y[2]^^z[2],linewidth(3));
draw(x[2]--x[0]^^y[2]--y[4]^^z[2]--z[7],linewidth(1));
label("A",(x[2].x,x[2].y-.3),S);
label("B",origin,S);
label("C",x[1],S);[/asy]
2022 Thailand TST, 2
Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.
2012 India Regional Mathematical Olympiad, 7
On the extension of chord $AB$ of a circle centroid at $O$ a point $X$ is taken and tangents $XC$ and $XD$ to the circle are drawn from it with $C$ and $D$ lying on the circle, let $E$ be the midpoint of the line segment $CD$. If $\angle OEB = 140^o$ then determine with proof the magnitude of $\angle AOB$.
2018 Moldova Team Selection Test, 3
Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.
2018 IMO Shortlist, G2
Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.
2019 Brazil Team Selection Test, 1
Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.
2004 IMO Shortlist, 4
In a convex quadrilateral $ABCD$, the diagonal $BD$ bisects neither the angle $ABC$ nor the angle $CDA$. The point $P$ lies inside $ABCD$ and satisfies \[\angle PBC=\angle DBA\quad\text{and}\quad \angle PDC=\angle BDA.\] Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP=CP$.
2007 District Olympiad, 1
Point $O$ is the intersection of the perpendicular bisectors of the sides of the triangle $\vartriangle ABC$ . Let $D$ be the intersection of the line $AO$ with the segment $[BC]$. Knowing that $OD = BD = \frac 13 BC$, find the measures of the angles of the triangle $\vartriangle ABC$.
1996 Bosnia and Herzegovina Team Selection Test, 3
Let $M$ be a point inside quadrilateral $ABCD$ such that $ABMD$ is parallelogram. If $\angle CBM = \angle CDM$ prove that $\angle ACD = \angle BCM$
2021 Macedonian Mathematical Olympiad, Problem 3
Let $ABCD$ be a trapezoid with $AD \parallel BC$ and $\angle BCD < \angle ABC < 90^\circ$. Let $E$ be the intersection point of the diagonals $AC$ and $BD$. The circumcircle $\omega$ of $\triangle BEC$ intersects the segment $CD$ at $X$. The lines $AX$ and $BC$ intersect at $Y$, while the lines $BX$ and $AD$ intersect at $Z$. Prove that the line $EZ$ is tangent to $\omega$ iff the line $BE$ is tangent to the circumcircle of $\triangle BXY$.
2011 Dutch IMO TST, 3
The circles $\Gamma_1$ and $\Gamma_2$ intersect at $D$ and $P$. The common tangent line of the two circles closest to point $D$ touches $\Gamma_1$ in A and $\Gamma_2$ in $B$. The line $AD$ intersects $\Gamma_2$ for the second time in $C$. Let $M$ be the midpoint of line segment $BC$. Prove that $\angle DPM = \angle BDC$.
2018 Hanoi Open Mathematics Competitions, 8
Let $ABCD$ be rhombus, with $\angle ABC = 80^o$: Let $E$ be midpoint of $BC$ and $F$ be perpendicular projection of $A$ onto $DE$. Find the measure of $\angle DFC$ in degree.
2018 Danube Mathematical Competition, 2
Let $ABC$ be a triangle such that in its interior there exists a point $D$ with $\angle DAC = \angle DCA = 30^o$ and $ \angle DBA = 60^o$. Denote $E$ the midpoint of the segment $BC$, and take $F$ on the segment $AC$ so that $AF = 2FC$. Prove that $DE \perp EF$.
2024 Bangladesh Mathematical Olympiad, P5
Let $I$ be the incenter of $\triangle ABC$ and $P$ be a point such that $PI$ is perpendicular to $BC$ and $PA$ is parallel to $BC$. Let the line parallel to $BC$, which is tangent to the incircle of $\triangle ABC$, intersect $AB$ and $AC$ at points $Q$ and $R$ respectively. Prove that $\angle BPQ = \angle CPR$.
2018 Dutch BxMO TST, 4
In a non-isosceles triangle $\vartriangle ABC$ we have $\angle BAC = 60^o$. Let $D$ be the intersection of the angular bisector of $\angle BAC$ with side $BC, O$ the centre of the circumcircle of $\vartriangle ABC$ and $E$ the intersection of $AO$ and $BC$. Prove that $\angle AED + \angle ADO = 90^o$.
2019 Macedonia National Olympiad, 3
Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.
2009 Greece Junior Math Olympiad, 2
From vertex $A$ of an equilateral triangle $ABC$, a ray $Ax$ intersects $BC$ at point $D$. Let $E$ be a point on $Ax$ such that $BA =BE$. Calculate $\angle AEC$.
1986 China Team Selection Test, 1
If $ABCD$ is a cyclic quadrilateral, then prove that the incenters of the triangles $ABC$, $BCD$, $CDA$, $DAB$ are the vertices of a rectangle.
2013 Peru MO (ONEM), 3
Let $P$ be a point inside the equilateral triangle $ABC$ such that $6\angle PBC = 3\angle PAC = 2\angle PCA$. Find the measure of the angle $\angle PBC$ .
2004 IMO, 1
1. Let $ABC$ be an acute-angled triangle with $AB\neq AC$. The circle with diameter $BC$ intersects the sides $AB$ and $AC$ at $M$ and $N$ respectively. Denote by $O$ the midpoint of the side $BC$. The bisectors of the angles $\angle BAC$ and $\angle MON$ intersect at $R$. Prove that the circumcircles of the triangles $BMR$ and $CNR$ have a common point lying on the side $BC$.
2020 Balkan MO Shortlist, G2
Let $G, H$ be the centroid and orthocentre of $\vartriangle ABC$ which has an obtuse angle at $\angle B$. Let $\omega$ be the circle with diameter $AG$. $\omega$ intersects $\odot(ABC)$ again at $L \ne A$. The tangent to $\omega$ at $L$ intersects $\odot(ABC)$ at $K \ne L$. Given that $AG = GH$, prove $\angle HKG = 90^o$
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[i]Sam Bealing, United Kingdom[/i]