Found problems: 1546
2010 Postal Coaching, 2
Suppose $\triangle ABC$ has circumcircle $\Gamma$, circumcentre $O$ and orthocentre $H$. Parallel lines $\alpha, \beta, \gamma$ are drawn through the vertices $A, B, C$, respectively. Let $\alpha ', \beta ', \gamma '$ be the reflections of $\alpha, \beta, \gamma$ in the sides $BC, CA, AB$, respectively.
$(a)$ Show that $\alpha ', \beta ', \gamma '$ are concurrent if and only if $\alpha, \beta, \gamma$ are parallel to the Euler line $OH$.
$(b)$ Suppose that $\alpha ', \beta ' , \gamma '$ are concurrent at the point $P$ . Show that $\Gamma$ bisects $OP$ .
2006 Federal Competition For Advanced Students, Part 1, 3
In the triangle $ ABC$ let $ D$ and $ E$ be the boundary points of the incircle with the sides $ BC$ and $ AC$. Show that if $ AD\equal{}BE$ holds, then the triangle is isoceles.
2009 German National Olympiad, 5
Let a triangle $ ABC$. $ E,F$ in segment $ AB$ so that $ E$ lie between $ AF$ and half of circle with diameter $ EF$ is tangent with $ BC,CA$ at $ G,H$. $ HF$ cut $ GE$ at $ S$, $ HE$ cut $ FG$ at $ T$. Prove that $ C$ is midpoint of $ ST$.
2000 France Team Selection Test, 2
$A,B,C,D$ are points on a circle in that order. Prove that $|AB-CD|+|AD-BC| \ge 2|AC-BD|$.
2014 Macedonia National Olympiad, 3
Let $k_1, k_2$ and $k_3$ be three circles with centers $O_1, O_2$ and $O_3$ respectively, such that no center is inside of the other two circles. Circles $k_1$ and $k_2$ intersect at $A$ and $P$, circles $k_1$ and $k_3$ intersect and $C$ and $P$, circles $k_2$ and $k_3$ intersect at $B$ and $P$. Let $X$ be a point on $k_1$ such that the line $XA$ intersects $k_2$ at $Y$ and the line $XC$ intersects $k_3$ at $Z$, such that $Y$ is nor inside $k_1$ nor inside $k_3$ and $Z$ is nor inside $k_1$ nor inside $k_2$.
a) Prove that $\triangle XYZ$ is simular to $\triangle O_1O_2O_3$
b) Prove that the $P_{\triangle XYZ} \le 4P_{\triangle O_1O_2O_3}$. Is it possible to reach equation?$
*Note: $P$ denotes the area of a triangle*
2009 Korea - Final Round, 4
$ABC$ is an acute triangle. (angle $C$ is bigger than angle $B$) Let $O$ be a center of the circle which passes $B$ and tangents to $AC$ at $C$. $O$ meets the segment $AB$ at $D$. $CO$ meets the circle $(O)$ again at $P$, a line, which passes $P$ and parallel to $AO$, meets $AC$ at $E$, and $EB$ meets the circle $(O)$ again at $L$. A perpendicular bisector of $BD$ meets $AC$ at $F$ and $LF$ meets $CD$ at $K$. Prove that two lines $EK$ and $CL$ are parallel.
2005 MOP Homework, 1
In convex hexagon $ ABCDEF$ all sides have equal length and
$ \angle A\plus{}\angle C\plus{}\angle E\equal{}\angle B\plus{}\angle D\plus{}\angle F$.
Prove that the diagonals $ AD,BE,CF$ are concurrent.
1977 Polish MO Finals, 1
Let $ABCD$ be a tetrahedron with $\angle BAD = 60^{\cdot}$, $\angle BAC = 40^{\cdot}$, $\angle ABD = 80^{\cdot}$, $\angle ABC = 70^{\cdot}$. Prove that the lines $AB$ and $CD$ are perpendicular.
2006 South africa National Olympiad, 2
Triangle $ABC$ has $BC=1$ and $AC=2$. What is the maximum possible value of $\hat{A}$.
2002 IberoAmerican, 3
Let $P$ be a point in the interior of the equilateral triangle $\triangle ABC$ such that $\sphericalangle{APC}=120^\circ$. Let $M$ be the intersection of $CP$ with $AB$, and $N$ the intersection of $AP$ and $BC$. Find the locus of the circumcentre of the triangle $MBN$ as $P$ varies.
2005 Regional Competition For Advanced Students, 2
Construct the semicircle $ h$ with the diameter $ AB$ and the midpoint $ M$. Now construct the semicircle $ k$ with the diameter $ MB$ on the same side as $ h$. Let $ X$ and $ Y$ be points on $ k$, such that the arc $ BX$ is $ \frac{3}{2}$ times the arc $ BY$. The line $ MY$ intersects the line $ BX$ in $ D$ and the semicircle $ h$ in $ C$.
Show that $ Y$ ist he midpoint of $ CD$.
2012 Sharygin Geometry Olympiad, 17
A square $ABCD$ is inscribed into a circle. Point $M$ lies on arc $BC$, $AM$ meets $BD$ in point $P$, $DM$ meets $AC$ in point $Q$. Prove that the area of quadrilateral $APQD$ is equal to the half of the area of the square.
2008 China Girls Math Olympiad, 5
In convex quadrilateral $ ABCD$, $ AB \equal{} BC$ and $ AD \equal{} DC$. Point $ E$ lies on segment $ AB$ and point $ F$ lies on segment $ AD$ such that $ B$, $ E$, $ F$, $ D$ lie on a circle. Point $ P$ is such that triangles $ DPE$ and $ ADC$ are similar and the corresponding vertices are in the same orientation (clockwise or counterclockwise). Point $ Q$ is such that triangles $ BQF$ and $ ABC$ are similar and the corresponding vertices are in the same orientation. Prove that points $ A$, $ P$, $ Q$ are collinear.
2005 MOP Homework, 7
Let $ABCD$ be a cyclic quadrilateral who interior angle at $B$ is $60$ degrees. Show that if $BC=CD$, then $CD+DA=AB$. Does the converse hold?
2014 Contests, 2
Suppose $ABCD$ is a convex quadrilateral.Points $P,Q,R$ and $S$ are four points on the line segments $AB,BC,CD$ and $DA$ respectively.The line segments $PR$ and $QS$ meet at $T$.Suppose that each of the quadrilaterals $APTS,BQTP,CRTQ$ and $DSTR$ have an incircle.Prove that the quadrilateral $ABCD$ also has an incircle.
2008 Hungary-Israel Binational, 3
P and Q are 2 points in the area bounded by 2 rays, e and f, coming out from a point O. Describe how to construct, with a ruler and a compass only, an isosceles triangle ABC, such that his base AB is on the ray e, the point C is on the ray f, P is on AC, and Q on BC.
2001 All-Russian Olympiad, 3
Points $A_1, B_1, C_1$ inside an acute-angled triangle $ABC$ are selected on the altitudes from $A, B, C$ respectively so that the sum of the areas of triangles $ABC_1, BCA_1$, and $CAB_1$ is equal to the area of triangle $ABC$. Prove that the circumcircle of triangle $A_1B_1C_1$ passes through the orthocenter $H$ of triangle $ABC$.
2010 Malaysia National Olympiad, 7
Let $ABC$ be a triangle in which $AB=AC$ and let $I$ be its incenter. It is known that $BC=AB+AI$. Let $D$ be a point on line $BA$ extended beyond $A$ such that $AD=AI$. Prove that $DAIC$ is a cyclic quadrilateral.
2014 Benelux, 4
Let $ABCD$ be a square. Consider a variable point $P$ inside the square for which $\angle BAP \ge 60^\circ.$ Let $Q$ be the intersection of the line $AD$ and the perpendicular to $BP$ in $P$. Let $R$ be the intersection of the line $BQ$ and the perpendicular to $BP$ from $C$.
[list]
[*] [b](a)[/b] Prove that $|BP|\ge |BR|$
[*] [b](b)[/b] For which point(s) $P$ does the inequality in [b](a)[/b] become an equality?[/list]
2012 Kazakhstan National Olympiad, 2
Given the rays $ OP$ and $OQ$.Inside the smaller angle $POQ$ selected points $M$ and $N$, such that $\angle POM=\angle QON $ and $\angle POM<\angle PON $ The circle, which concern the rays $OP$ and $ON$, intersects the second circle, which concern the rays $OM$ and $OQ$ at the points $B$ and $C$. Prove that$\angle POC=\angle QOB $
2011 Morocco National Olympiad, 4
Two circles $C_{1}$ and $C_{2}$ intersect in $A$ and $B$. A line passing through $B$ intersects $C_{1}$ in $C$ and $C_{2}$ in $D$. Another line passing through $B$ intersects $C_{1}$ in $E$ and $C_{2}$ in $F$, $(CF)$ intersects $C_{1}$ and $C_{2}$ in $P$ and $Q$ respectively. Make sure that in your diagram, $B, E, C, A, P \in C_{1}$ and $B, D, F, A, Q \in C_{2}$ in this order. Let $M$ and $N$ be the middles of the arcs $BP$ and $BQ$ respectively. Prove that if $CD=EF$, then the points $C,F,M,N$ are cocylic in this order.
1990 Vietnam National Olympiad, 3
A tetrahedron is to be cut by three planes which form a parallelepiped whose three faces and all vertices lie on the surface of the tetrahedron.
(a) Can this be done so that the volume of the parallelepiped is at least $ \frac{9}{40}$ of the volume of the tetrahedron?
(b) Determine the common point of the three planes if the volume of the parallelepiped is $ \frac{11}{50}$ of the volume of the tetrahedron.
Indonesia MO Shortlist - geometry, g2.6
Let $ABC$ be a triangle. Suppose $D$ is on $BC$ such that $AD$ bisects $\angle BAC$. Suppose $M$ is on $AB$ such that $\angle MDA = \angle ABC$, and $N$ is on $AC$ such that $\angle NDA = \angle ACB$. If $AD$ and $MN$ intersect on $P$, prove that $AD^3 = AB \cdot AC \cdot AP$.
1972 IMO Longlists, 17
A solid right circular cylinder with height $h$ and base-radius $r$ has a solid hemisphere of radius $r$ resting upon it. The center of the hemisphere $O$ is on the axis of the cylinder. Let $P$ be any point on the surface of the hemisphere and $Q$ the point on the base circle of the cylinder that is furthest from $P$ (measuring along the surface of the combined solid). A string is stretched over the surface from $P$ to $Q$ so as to be as short as possible. Show that if the string is not in a plane, the straight line $PO$ when produced cuts the curved surface of the cylinder.
2007 Croatia Team Selection Test, 3
Let $ABC$ be a triangle such that $|AC|>|AB|$. Let $X$ be on line $AB$ (closer to $A$) such that $|BX|=|AC|$ and let $Y$ be on the segment $AC$ such that $|CY|=|AB|$. Intersection of lines $XY$ and bisector of $BC$ is point $P$. Prove that $\angle BPC+\angle BAC = 180^\circ$.