Found problems: 1546
1988 Polish MO Finals, 3
$W$ is a polygon which has a center of symmetry $S$ such that if $P$ belongs to $W$, then so does $P'$, where $S$ is the midpoint of $PP'$. Show that there is a parallelogram $V$ containing $W$ such that the midpoint of each side of $V$ lies on the border of $W$.
1995 APMO, 4
Let $C$ be a circle with radius $R$ and centre $O$, and $S$ a fixed point in the interior of $C$. Let $AA'$ and $BB'$ be perpendicular chords through $S$. Consider the rectangles $SAMB$, $SBN'A'$, $SA'M'B'$, and $SB'NA$. Find the set of all points $M$, $N'$, $M'$, and $N$ when $A$ moves around the whole circle.
2001 Korea - Final Round, 2
Let $P$ be a given point inside a convex quadrilateral $O_1O_2O_3O_4$. For each $i = 1,2,3,4$, consider the lines $l$ that pass through $P$ and meet the rays $O_iO_{i-1}$ and $O_iO_{i+1}$ (where $O_0 = O_4$ and $O_5 = O_1$) at distinct points $A_i(l)$ and $B_i(l)$, respectively. Denote $f_i(l) = PA_i(l) \cdot PB_i(l)$. Among all such lines $l$, let $l_i$ be the one that minimizes $f_i$. Show that if $l_1 = l_3$ and $l_2 = l_4$, then the quadrilateral $O_1O_2O_3O_4$ is a parallelogram.
2008 Sharygin Geometry Olympiad, 7
(F.Nilov) Given isosceles triangle $ ABC$ with base $ AC$ and $ \angle B \equal{} \alpha$. The arc $ AC$ constructed outside the triangle has angular measure equal to $ \beta$. Two lines passing through $ B$ divide the segment and the arc $ AC$ into three equal parts. Find the ratio $ \alpha / \beta$.
2003 France Team Selection Test, 3
$M$ is an arbitrary point inside $\triangle ABC$. $AM$ intersects the circumcircle of the triangle again at $A_1$. Find the points $M$ that minimise $\frac{MB\cdot MC}{MA_1}$.
2008 Germany Team Selection Test, 2
Let $ ABCD$ be an isosceles trapezium with $ AB \parallel{} CD$ and $ \bar{BC} \equal{} \bar{AD}.$ The parallel to $ AD$ through $ B$ meets the perpendicular to $ AD$ through $ D$ in point $ X.$ The line through $ A$ drawn which is parallel to $ BD$ meets the perpendicular to $ BD$ through $ D$ in point $ Y.$ Prove that points $ C,X,D$ and $ Y$ lie on a common circle.
1962 Vietnam National Olympiad, 3
Let $ ABCD$ is a tetrahedron. Denote by $ A'$, $ B'$ the feet of the perpendiculars from $ A$ and $ B$, respectively to the opposite faces. Show that $ AA'$ and $ BB'$ intersect if and only if $ AB$ is perpendicular to $ CD$. Do they intersect if $ AC \equal{} AD \equal{} BC \equal{} BD$?
2006 Estonia National Olympiad, 4
Let O be the circumcentre of an acute triangle ABC and let A′, B′ and C′ be the
circumcentres of triangles BCO, CAO and ABO, respectively. Prove that the area of triangle ABC does not exceed the area of triangle A′B′C′.
1999 Mediterranean Mathematics Olympiad, 4
In triangle $\triangle ABC$ we have $BC=a,CA=b,AB=c$ and $\angle B=4\angle A$ Show that \[ab^2c^3=(b^2-a^2-ac)((a^2-b^2)^2-a^2c^2)\]
2014 Vietnam National Olympiad, 4
Let $ABC$ be an acute triangle, $(O)$ be the circumcircle, and $AB<AC.$ Let $I$ be the midpoint of arc $BC$ (not containing $A$). $K$ lies on $AC,$ $K\ne C$ such that $IK=IC.$ $BK$ intersects $(O)$ at the second point $D,$ $D\ne B$ and intersects $AI$ at $E.$ $DI$ intersects $AC$ at $F.$
a) Prove that $EF=\frac{BC}{2}.$
b) $M$ lies on $DI$ such that $CM$ is parallel to $AD.$ $KM$ intersects $BC$ at $N.$ The circumcircle of triangle $BKN$ intersects $(O)$ at the second point $P.$ Prove that $PK$ passes through the midpoint of segment $AD.$
2009 Ukraine National Mathematical Olympiad, 3
Point $O$ is inside triangle $ABC$ such that $\angle AOB = \angle BOC = \angle COA = 120^\circ .$ Prove that
\[\frac{AO^2}{BC}+\frac{BO^2}{CA}+\frac{CO^2}{AB} \geq \frac{AO+BO+CO}{\sqrt 3}.\]
1978 IMO Longlists, 38
Given a circle, construct a chord that is trisected by two given noncollinear radii.
2014 India IMO Training Camp, 3
In a triangle $ABC$, points $X$ and $Y$ are on $BC$ and $CA$ respectively such that $CX=CY$,$AX$ is not perpendicular to $BC$ and $BY$ is not perpendicular to $CA$.Let $\Gamma$ be the circle with $C$ as centre and $CX$ as its radius.Find the angles of triangle $ABC$ given that the orthocentres of triangles $AXB$ and $AYB$ lie on $\Gamma$.
2014 Contests, 3
In obtuse triangle $ABC$, with the obtuse angle at $A$, let $D$, $E$, $F$ be the feet of the altitudes through $A$, $B$, $C$ respectively. $DE$ is parallel to $CF$, and $DF$ is parallel to the angle bisector of $\angle BAC$. Find the angles of the triangle.
2011 German National Olympiad, 3
Let $ABC$ be an acute triangle and $D$ the foot of the altitude from $A$ onto $BC$. A semicircle with diameter $BC$ intersects segments $AB,AC$ and $AD$ in the points $F,E$ resp. $X$. The circumcircles of the triangles $DEX$ and $DXF$ intersect $BC$ in $L$ resp. $N$ other than $D$. Prove $BN=LC$.
2011 Croatia Team Selection Test, 3
Let $K$ and $L$ be the points on the semicircle with diameter $AB$. Denote intersection of $AK$ and $AL$ as $T$ and let $N$ be the point such that $N$ is on segment $AB$ and line $TN$ is perpendicular to $AB$. If $U$ is the intersection of perpendicular bisector of $AB$ an $KL$ and $V$ is a point on $KL$ such that angles $UAV$ and $UBV$ are equal. Prove that $NV$ is perpendicular to $KL$.
1999 All-Russian Olympiad, 3
A circle touches sides $DA$, $AB$, $BC$, $CD$ of a quadrilateral $ABCD$ at points $K$, $L$, $M$, $N$, respectively. Let $S_1$, $S_2$, $S_3$, $S_4$ respectively be the incircles of triangles $AKL$, $BLM$, $CMN$, $DNK$. The external common tangents distinct from the sides of $ABCD$ are drawn to $S_1$ and $S_2$, $S_2$ and $S_3$, $S_3$ and $S_4$, $S_4$ and $S_1$. Prove that these four tangents determine a rhombus.
May Olympiad L2 - geometry, 2012.3
Given Triangle $ABC$, $\angle B= 2 \angle C$, and $\angle A>90^\circ$. Let $M$ be midpoint of $BC$. Perpendicular of $AC$ at $C$ intersects $AB$ at $D$. Show $\angle AMB = \angle DMC$
[hide]If possible, don't use projective geometry[/hide]
2008 Tuymaada Olympiad, 8
A convex hexagon is given. Let $ s$ be the sum of the lengths of the three segments connecting the midpoints of its opposite sides. Prove that there is a point in the hexagon such that the sum of its distances to the lines containing the sides of the hexagon does not exceed $ s.$
[i]Author: N. Sedrakyan[/i]
1988 Kurschak Competition, 3
Consider the convex lattice quadrilateral $PQRS$ whose diagonals intersect at $E$. Prove that if $\angle P+\angle Q<180^\circ$, then the $\triangle PQE$ contains inside it or on one of its sides a lattice point other than $P$ and $Q$.
2013 Iran MO (3rd Round), 2
Let $ABC$ be a triangle with circumcircle $(O)$. Let $M,N$ be the midpoint of arc $AB,AC$ which does not contain $C,B$ and let $M',N'$ be the point of tangency of incircle of $\triangle ABC$ with $AB,AC$. Suppose that $X,Y$ are foot of perpendicular of $A$ to $MM',NN'$. If $I$ is the incenter of $\triangle ABC$ then prove that quadrilateral $AXIY$ is cyclic if and only if $b+c=2a$.
2008 South africa National Olympiad, 2
Let $ABCD$ be a convex quadrilateral with the property that $AB$ extended and $CD$ extended intersect at a right angle. Prove that $AC\cdot BD>AD\cdot BC$.
1986 Vietnam National Olympiad, 1
Let $ ABCD$ be a square of side $ 2a$. An equilateral triangle $ AMB$ is constructed in the plane through $ AB$ perpendicular to the plane of the square. A point $ S$ moves on $ AB$ such that $ SB\equal{}x$. Let $ P$ be the projection of $ M$ on $ SC$ and $ E$, $ O$ be the midpoints of $ AB$ and $ CM$ respectively.
(a) Find the locus of $ P$ as $ S$ moves on $ AB$.
(b) Find the maximum and minimum lengths of $ SO$.
2002 Bulgaria National Olympiad, 2
Consider the orthogonal projections of the vertices $A$, $B$ and $C$ of triangle $ABC$ on external bisectors of $ \angle ACB$, $ \angle BAC$ and $ \angle ABC$, respectively. Prove that if $d$ is the diameter of the circumcircle of the triangle, which is formed by the feet of projections, while $r$ and $p$ are the inradius and the semiperimeter of triangle $ABC$, prove that $r^2+p^2=d^2$
[i]Proposed by Alexander Ivanov[/i]
2008 Silk Road, 2
In a triangle $ABC$ $A_0$,$B_0$ and $C_0$ are the midpoints of the sides $BC$,$CA$ and $AB$.$A_1$,$B_1$,$C_1$ are the midpoints of the broken lines $BAC,CAB,ABC$.Show that $A_0A_1,B_0B_1,C_0C_1$ are concurrent.