Found problems: 1546
2011 Morocco National Olympiad, 4
$ (C)$ and $(C')$ are two circles which intersect in $A$ and $B$. $(D)$ is a line that moves and passes through $A$, intersecting $(C)$ in P and $(C')$ in P'.
Prove that the bisector of $[PP']$ passes through a non-moving point.
2014 Romania Team Selection Test, 1
Let $ABC$ be a triangle and let $X$,$Y$,$Z$ be interior points on the sides $BC$, $CA$, $AB$, respectively. Show that the magnified image of the triangle $XYZ$ under a homothety of factor $4$ from its centroid covers at least one of the vertices $A$, $B$, $C$.
2010 China Team Selection Test, 1
Let $\triangle ABC$ be an acute triangle with $AB>AC$, let $I$ be the center of the incircle. Let $M,N$ be the midpoint of $AC$ and $AB$ respectively. $D,E$ are on $AC$ and $AB$ respectively such that $BD\parallel IM$ and $CE\parallel IN$. A line through $I$ parallel to $DE$ intersects $BC$ in $P$. Let $Q$ be the projection of $P$ on line $AI$. Prove that $Q$ is on the circumcircle of $\triangle ABC$.
2005 Germany Team Selection Test, 2
Let $M$ be a set of points in the Cartesian plane, and let $\left(S\right)$ be a set of segments (whose endpoints not necessarily have to belong to $M$) such that one can walk from any point of $M$ to any other point of $M$ by travelling along segments which are in $\left(S\right)$. Find the smallest total length of the segments of $\left(S\right)$ in the cases
[b]a.)[/b] $M = \left\{\left(-1,0\right),\left(0,0\right),\left(1,0\right),\left(0,-1\right),\left(0,1\right)\right\}$.
[b]b.)[/b] $M = \left\{\left(-1,-1\right),\left(-1,0\right),\left(-1,1\right),\left(0,-1\right),\left(0,0\right),\left(0,1\right),\left(1,-1\right),\left(1,0\right),\left(1,1\right)\right\}$.
In other words, find the Steiner trees of the set $M$ in the above two cases.
2016 Bundeswettbewerb Mathematik, 3
Let $A,B,C$ and $D$ be points on a circle in this order. The chords $AC$ and $BD$ intersect in point $P$. The perpendicular to $AC$ through C and the perpendicular to $BD$ through $D$ intersect in point $Q$.
Prove that the lines $AB$ and $PQ$ are perpendicular.
2012 Sharygin Geometry Olympiad, 18
A triangle and two points inside it are marked. It is known that one of the triangle’s angles is equal to $58^{\circ}$, one of two remaining angles is equal to $59^{\circ}$, one of two given points is the incenter of the triangle and the second one is its circumcenter. Using only the ruler without partitions determine where is each of the angles and where is each of the centers.
2013 Pan African, 1
Let $ABCD$ be a convex quadrilateral with $AB$ parallel to $CD$. Let $P$ and $Q$ be the midpoints of $AC$ and $BD$, respectively. Prove that if $\angle ABP=\angle CBD$, then $\angle BCQ=\angle ACD$.
1974 IMO Longlists, 5
A straight cone is given inside a rectangular parallelepiped $B$, with the apex at one of the vertices, say $T$ , of the parallelepiped, and the base touching the three faces opposite to $T .$ Its axis lies at the long diagonal through $T.$ If $V_1$ and $V_2$ are the volumes of the cone and the parallelepiped respectively, prove that
\[V_1 \leq \frac{\sqrt 3 \pi V_2}{27}.\]
1991 China Team Selection Test, 1
We choose 5 points $A_1, A_2, \ldots, A_5$ on a circumference of radius 1 and centre $O.$ $P$ is a point inside the circle. Denote $Q_i$ as the intersection of $A_iA_{i+2}$ and $A_{i+1}P$, where $A_7 = A_2$ and $A_6 = A_1.$ Let $OQ_i = d_i, i = 1,2, \ldots, 5.$ Find the product $\prod^5_{i=1} A_iQ_i$ in terms of $d_i.$
1988 Irish Math Olympiad, 2
A; B; C; D are the vertices of a square, and P is a point on the arc CD of
its circumcircle. Prove that
$ |PA|^2 - |PB|^2 = |PB|.|PD| -|PA|.|PC| $
Can anyone here find the solution? I'm not great with geometry, so i tried turning it into co-ordinate geometry equations, but sadly to no avail. Thanks in advance.
2009 Tuymaada Olympiad, 3
In a cyclic quadrilateral $ ABCD$ the sides $ AB$ and $ AD$ are equal, $ CD>AB\plus{}BC$. Prove that $ \angle ABC>120^\circ$.
2001 Vietnam National Olympiad, 1
A circle center $O$ meets a circle center $O'$ at $A$ and $B.$ The line $TT'$ touches the first circle at $T$ and the second at $T'$. The perpendiculars from $T$ and $T'$ meet the line $OO'$ at $S$ and $S'$. The ray $AS$ meets the first circle again at $R$, and the ray $AS'$ meets the second circle again at $R'$. Show that $R, B$ and $R'$ are collinear.
2010 Contests, 1
$ABC$ is an acute angle triangle such that $AB>AC$ and $\hat{BAC}=60^{\circ}$. Let's denote by $O$ the center of the circumscribed circle of the triangle and $H$ the intersection of altitudes of this triangle. Line $OH$ intersects $AB$ in point $P$ and $AC$ in point $Q$. Find the value of the ration $\frac{PO}{HQ}$.
2012 Indonesia TST, 3
Suppose $l(M, XYZ)$ is a Simson line of the triangle $XYZ$ that passes through $M$.
Suppose $ABCDEF$ is a cyclic hexagon such that $l(A, BDF), l(B, ACE), l(D, ABF), l(E, ABC)$ intersect at a single point. Prove that $CDEF$ is a rectangle.
[color=blue]Should the first sentence read:
Suppose $l(M, XYZ)$ is a Simson line of the triangle $XYZ$ [u]with respect to[/u] $M$.
? Since it appears weird that a Simson line that passes a point is to be constructed. However, this is Unsolved after all, so I'm not sure.[/color]
2000 Finnish National High School Mathematics Competition, 1
Two circles are externally tangent at the point $A$. A common tangent of the circles meets one circle at the point $B$ and another at the point $C$ ($B \ne C)$. Line segments $BD$ and $CE$ are diameters of the circles. Prove that the points $D, A$ and $C$ are collinear.
2009 Tournament Of Towns, 4
A point is chosen on each side of a regular $2009$-gon. Let $S$ be the area of the $2009$-gon with vertices at these points. For each of the chosen points, reflect it across the midpoint of its side. Prove that the $2009$-gon with vertices at the images of these reflections also has area $S.$
[i](4 points)[/i]
2001 Tournament Of Towns, 4
[b]a.[/b] There are $5$ identical paper triangles on the table. Each can be moved in any direction parallel to itself (i.e., without rotating it). Is it true that then any one of them can be covered by the $4$ others?
[b]b.[/b] There are $5$ identical equilateral paper triangles on the table. Each can be moved in any direction parallel to itself. Prove that any one of them can be covered by the $4$ others in this way.
2008 Serbia National Math Olympiad, 2
Triangle $ \triangle ABC$ is given. Points $ D$ i $ E$ are on line $ AB$ such that $ D \minus{} A \minus{} B \minus{} E, AD \equal{} AC$ and $ BE \equal{} BC$. Bisector of internal angles at $ A$ and $ B$ intersect $ BC,AC$ at $ P$ and $ Q$, and circumcircle of $ ABC$ at $ M$ and $ N$. Line which connects $ A$ with center of circumcircle of $ BME$ and line which connects $ B$ and center of circumcircle of $ AND$ intersect at $ X$. Prove that $ CX \perp PQ$.
2002 China Team Selection Test, 2
There are $ n$ points ($ n \geq 4$) on a sphere with radius $ R$, and not all of them lie on the same semi-sphere. Prove that among all the angles formed by any two of the $ n$ points and the sphere centre $ O$ ($ O$ is the vertex of the angle), there is at least one that is not less than $ \displaystyle 2 \arcsin{\frac{\sqrt{6}}{3}}$.
2007 Mediterranean Mathematics Olympiad, 2
The diagonals $AC$ and $BD$ of a convex cyclic quadrilateral $ABCD$ intersect at point $E$. Given that $AB = 39, AE = 45, AD = 60$ and $BC = 56$, determine the length of $CD.$
1974 IMO Longlists, 17
Show that there exists a set $S$ of $15$ distinct circles on the surface of a sphere, all having the same radius and such that $5$ touch exactly $5$ others, $5$ touch exactly $4$ others, and $5$ touch exactly $3$ others.
[i][General Problem: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=384764][/i]
2006 South East Mathematical Olympiad, 2
In $\triangle ABC$, $\angle ABC=90^{\circ}$. Points $D,G$ lie on side $AC$. Points $E, F$ lie on segment $BD$, such that $AE \perp BD $ and $GF \perp BD$. Show that if $BE=EF$, then $\angle ABG=\angle DFC$.
2013 Hong kong National Olympiad, 3
Let $ABC$ be a triangle with $CA>BC>AB$. Let $O$ and $H$ be the circumcentre and orthocentre of triangle $ABC$ respectively. Denote by $D$ and $E$ the midpoints of the arcs $AB$ and $AC$ of the circumcircle of triangle $ABC$ not containing the opposite vertices. Let $D'$ be the reflection of $D$ about $AB$ and $E'$ the reflection of $E$ about $AC$. Prove that $O,H,D',E'$ are concylic if and only if $A,D',E'$ are collinear.
1998 South africa National Olympiad, 3
$A,\ B,\ C,\ D,\ E$ and $F$ lie (in that order) on the circumference of a circle. The chords $AD,\ BE$ and $CF$ are concurrent. $P,\ Q$ and $R$ are the midpoints of $AD,\ BE$ and $CF$ respectively. Two further chords $AG \parallel BE$ and $AH \parallel CF$ are drawn. Show that $PQR$ is similar to $DGH$.
1997 Romania Team Selection Test, 4
Let $w$ be a circle and $AB$ a line not intersecting $w$. Given a point $P_{0}$ on $w$, define the sequence $P_{0},P_{1},\ldots $ as follows: $P_{n\plus{}1}$ is the second intersection with $w$ of the line passing through $B$ and the second intersection of the line $AP_{n}$ with $w$. Prove that for a positive integer $k$, if $P_{0}\equal{}P_{k}$ for some choice of $P_{0}$, then $P_{0}\equal{}P_{k}$ for any choice of $P_{0}$.
[i]Gheorge Eckstein[/i]