Olympiad problems

2006 Turkey MO (2nd round), 1

Tags: geometry , circumcircle , power of a point , radical axis , cyclic quadrilateral , geometry unsolved

Points $P$ and $Q$ on side $AB$ of a convex quadrilateral $ABCD$ are given such that $AP = BQ.$ The circumcircles of triangles $APD$ and $BQD$ meet again at $K$ and those of $APC$ and $BQC$ meet again at $L$. Show that the points $D,C,K,L$ lie on a circle.

2005 IMAR Test, 3

Tags: geometry unsolved , geometry

A convex polygon is given, no two of whose sides are parallel. For each side we consider the angle the side subtends at the vertex farthest from the side. Show that the sum of these angles equals 180 degrees.

2009 Greece National Olympiad, 2

Tags: geometry , circumcircle , geometric transformation , homothety , geometry unsolved

Consider a triangle $ABC$ with circumcenter $O$ and let $A_1,B_1,C_1$ be the midpoints of the sides $BC,AC,AB,$ respectively. Points $A_2,B_2,C_2$ are defined as $\overrightarrow{OA_2}=\lambda \cdot \overrightarrow{OA_1}, \ \overrightarrow{OB_2}=\lambda \cdot \overrightarrow{OB_1}, \ \overrightarrow{OC_2}=\lambda \cdot \overrightarrow{OC_1},$ where $\lambda >0.$ Prove that lines $AA_2,BB_2,CC_2$ are concurrent.

2004 Greece National Olympiad, 3

Tags: symmetry , geometry unsolved , geometry

Consider a circle $K(O,r)$ and a point $A$ outside $K.$ A line $\epsilon$ different from $AO$ cuts $K$ at $B$ and $C,$ where $B$ lies between $A$ and $C.$ Now the symmetric line of $\epsilon$ with respect to axis of symmetry the line $AO$ cuts $K$ at $E$ and $D,$ where $E$ lies between $A$ and $D.$ Show that the diagonals of the quadrilateral $BCDE$ intersect in a fixed point.

1983 IMO Longlists, 34

Tags: geometry unsolved , geometry

In a plane are given n points $P_i \ (i = 1, 2, \ldots , n)$ and two angles $\alpha$ and $\beta$. Over each of the segments $P_iP_{i+1} \ (P_{n+1} = P_1)$ a point $Q_i$ is constructed such that for all $i$: [b](i)[/b] upon moving from $P_i$ to $P_{i+1}, Q_i$ is seen on the same side of $P_iP_{i+1}$, [b](ii)[/b] $\angle P_{i+1}P_iQ_i = \alpha,$ [b](iii)[/b] $\angle P_iP_{i+1}Q_i = \beta.$ Furthermore, let $g$ be a line in the same plane with the property that all the points $P_i,Q_i$ lie on the same side of $g$. Prove that \[\sum_{i=1}^n d(P_i, g)= \sum_{i=1}^n d(Q_i, g).\] where $d(M,g)$ denotes the distance from point $M$ to line $g.$

1988 Austrian-Polish Competition, 3

Tags: geometry , rhombus , parallelogram , cyclic quadrilateral , geometry unsolved

In a ABCD cyclic quadrilateral 4 points K, L ,M, N are taken on AB , BC , CD and DA , respectively such that KLMN is a parallelogram. Lines AD, BC and KM have a common point. And also lines AB, DC and NL have a common point. Prove that KLMN is rhombus.

2007 Vietnam National Olympiad, 3

Tags: geometry , trapezoid , geometry unsolved

Let ABCD be trapezium that is inscribed in circle (O) with larger edge BC. P is a point lying outer segment BC. PA cut (O) at N(that means PA isn't tangent of (O)), the circle with diameter PD intersect (O) at E, DE meet BC at N. Prove that MN always pass through a fixed point.

2008 India Regional Mathematical Olympiad, 1

Tags: geometry , circumcircle , geometric transformation , reflection , rectangle , trigonometry , geometry unsolved

Let $ ABC$ be an acute angled triangle; let $ D,F$ be the midpoints of $ BC,AB$ respectively. Let the perpendicular from $ F$ to $ AC$ and the perpendicular from $ B$ ti $ BC$ meet in $ N$: Prove that $ ND$ is the circumradius of $ ABC$. [15 points out of 100 for the 6 problems]

2013 Romania Team Selection Test, 2

Tags: geometry unsolved , geometry

Let $\gamma$ a circle and $P$ a point who lies outside the circle. Two arbitrary lines $l$ and $l'$ which pass through $P$ intersect the circle at the points $X$, $Y$ , respectively $X'$, $Y'$ , such that $X$ lies between $P$ and $Y$ and $X'$ lies between $P$ and $Y'$. Prove that the line determined by the circumcentres of the triangles $PXY'$ and $PX'Y$ passes through a fixed point.

2009 Germany Team Selection Test, 3

Tags: geometry , parallelogram , geometry unsolved

Let $ A,B,C,M$ points in the plane and no three of them are on a line. And let $ A',B',C'$ points such that $ MAC'B, MBA'C$ and $ MCB'A$ are parallelograms: (a) Show that \[ \overline{MA} \plus{} \overline{MB} \plus{} \overline{MC} < \overline{AA'} \plus{} \overline{BB'} \plus{} \overline{CC'}.\] (b) Assume segments $ AA', BB'$ and $ CC'$ have the same length. Show that $ 2 \left(\overline{MA} \plus{} \overline{MB} \plus{} \overline{MC} \right) \leq \overline{AA'} \plus{} \overline{BB'} \plus{} \overline{CC'}.$ When do we have equality?

1977 IMO Longlists, 33

Tags: vector , analytic geometry , geometry unsolved , geometry

A circle $K$ centered at $(0,0)$ is given. Prove that for every vector $(a_1,a_2)$ there is a positive integer $n$ such that the circle $K$ translated by the vector $n(a_1,a_2)$ contains a lattice point (i.e., a point both of whose coordinates are integers).

1976 IMO Longlists, 49

Tags: trigonometry , geometry unsolved , geometry

Determine whether there exist $1976$ nonsimilar triangles with angles $\alpha, \beta, \gamma,$ each of them satisfying the relations \[\frac{\sin \alpha + \sin\beta + \sin\gamma}{\cos \alpha + \cos \beta + \cos \gamma}=\frac{12}{7}\text{ and }\sin \alpha \sin \beta \sin \gamma =\frac{12}{25}\]

1978 IMO Longlists, 42

Tags: geometry unsolved , geometry

$A,B,C,D,E$ are points on a circle $O$ with radius equal to $r$. Chords $AB$ and $DE$ are parallel to each other and have length equal to $x$. Diagonals $AC,AD,BE, CE$ are drawn. If segment $XY$ on $O$ meets $AC$ at $X$ and $EC$ at $Y$ , prove that lines $BX$ and $DY$ meet at $Z$ on the circle.

1993 Irish Math Olympiad, 1

Tags: analytic geometry , modular arithmetic , geometry unsolved , geometry

Show that among any five points $ P_1,...,P_5$ with integer coordinates in the plane, there exists at least one pair $ (P_i,P_j)$, with $ i \not\equal{} j$ such that the segment $ P_i P_j$ contains a point $ Q$ with integer coordinates other than $ P_i, P_j$.

2012 Sharygin Geometry Olympiad, 3

Tags: geometry , incenter , perpendicular bisector , angle bisector , geometry unsolved

A circle with center $I$ touches sides $AB,BC,CA$ of triangle $ABC$ in points $C_{1},A_{1},B_{1}$. Lines $AI, CI, B_{1}I$ meet $A_{1}C_{1}$ in points $X, Y, Z$ respectively. Prove that $\angle Y B_{1}Z = \angle XB_{1}Z$.

2015 Belarus Team Selection Test, 2

Tags: geometry , circumcircle , geometric transformation , reflection , cyclic quadrilateral , geometry unsolved

In a cyclic quadrilateral $ABCD$, the extensions of sides $AB$ and $CD$ meet at point $P$, and the extensions of sides $AD$ and $BC$ meet at point $Q$. Prove that the distance between the orthocenters of triangles $APD$ and $AQB$ is equal to the distance between the orthocenters of triangles $CQD$ and $BPC$.

2006 MOP Homework, 4

Tags: geometry , parallelogram , Ross Mathematics Program , geometry unsolved

1.14. Let P and Q be interior points of triangle ABC such that \ACP = \BCQ and \CAP = \BAQ. Denote by D;E and F the feet of the perpendiculars from P to the lines BC, CA and AB, respectively. Prove that if \DEF = 90, then Q is the orthocenter of triangle BDF.

1994 Dutch Mathematical Olympiad, 4

Tags: geometry , rectangle , geometry unsolved

Let $ P$ be a point on the diagonal $ BD$ of a rectangle $ ABCD$, $ F$ be the projection of $ P$ on $ BC$, and $ H \not\equal{} B$ be the point on $ BC$ such that $ BF\equal{}FH$. If lines $ PC$ and $ AH$ intersect at $ Q$, prove that the areas of triangles $ APQ$ and $ CHQ$ are equal.

2008 Ukraine Team Selection Test, 9

Tags: geometry , parallelogram , ratio , geometric transformation , homothety , geometry unsolved

Given $ \triangle ABC$ with point $ D$ inside. Let $ A_0\equal{}AD\cap BC$, $ B_0\equal{}BD\cap AC$, $ C_0 \equal{}CD\cap AB$ and $ A_1$, $ B_1$, $ C_1$, $ A_2$, $ B_2$, $ C_2$ are midpoints of $ BC$, $ AC$, $ AB$, $ AD$, $ BD$, $ CD$ respectively. Two lines parallel to $ A_1A_2$ and $ C_1C_2$ and passes through point $ B_0$ intersects $ B_1B_2$ in points $ A_3$ and $ C_3$respectively. Prove that $ \frac{A_3B_1}{A_3B_2}\equal{}\frac{C_3B_1}{C_3B_2}$.

2003 Federal Competition For Advanced Students, Part 2, 3

Tags: projective geometry , geometry unsolved , geometry

Let $ABC$ be an acute-angled triangle. The circle $k$ with diameter $AB$ intersects $AC$ and $BC$ again at $P$ and $Q$, respectively. The tangents to $k$ at $A$ and $Q$ meet at $R$, and the tangents at $B$ and $P$ meet at $S$. Show that $C$ lies on the line $RS$.

2010 Tuymaada Olympiad, 3

Tags: geometry , circumcircle , geometric transformation , reflection , cyclic quadrilateral , geometry unsolved

In a cyclic quadrilateral $ABCD$, the extensions of sides $AB$ and $CD$ meet at point $P$, and the extensions of sides $AD$ and $BC$ meet at point $Q$. Prove that the distance between the orthocenters of triangles $APD$ and $AQB$ is equal to the distance between the orthocenters of triangles $CQD$ and $BPC$.

1999 Canada National Olympiad, 2

Tags: geometry unsolved , geometry

Let $ABC$ be an equilateral triangle of altitude 1. A circle with radius 1 and center on the same side of $AB$ as $C$ rolls along the segment $AB$. Prove that the arc of the circle that is inside the triangle always has the same length.

2008 Ukraine Team Selection Test, 4

Tags: geometry , geometry unsolved

Two circles $ \omega_1$ and $ \omega_2$ tangents internally in point $ P$. On their common tangent points $ A$, $ B$ are chosen such that $ P$ lies between $ A$ and $ B$. Let $ C$ and $ D$ be the intersection points of tangent from $ A$ to $ \omega_1$, tangent from $ B$ to $ \omega_2$ and tangent from $ A$ to $ \omega_2$, tangent from $ B$ to $ \omega_1$, respectively. Prove that $ CA \plus{} CB \equal{} DA \plus{} DB$.

2010 Contests, 2

Tags: geometry unsolved , geometry

Karlson and Smidge divide a cake in a shape of a square in the following way. First, Karlson places a candle on the cake (chooses some interior point). Then Smidge makes a straight cut from the candle to the boundary in the direction of his choice. Then Karlson makes a straight cut from the candle to the boundary in the direction perpendicular to Smidge's cut. As a result, the cake is split into two pieces; Smidge gets the smaller one. Smidge wants to get a piece which is no less than a quarter of the cake. Can Karlson prevent Smidge from getting the piece of that size?

2007 Brazil National Olympiad, 5

Tags: geometry unsolved , geometry

Let $ ABCD$ be a convex quadrangle, $ P$ the intersection of lines $ AB$ and $ CD$, $ Q$ the intersection of lines $ AD$ and $ BC$ and $ O$ the intersection of diagonals $ AC$ and $ BD$. Show that if $ \angle POQ\equal{} 90^\circ$ then $ PO$ is the bisector of $ \angle AOD$ and $ OQ$ is the bisector of $ \angle AOB$.