Olympiad problems

1995 All-Russian Olympiad, 2

A chord $CD$ of a circle with center $O$ is perpendicular to a diameter $AB$. A chord $AE$ bisects the radius $OC$. Show that the line $DE$ bisects the chord $BC$ [i]V. Gordon[/i]

2003 Flanders Math Olympiad, 2

Tags: geometry proposed , geometry

Two circles $C_1$ and $C_2$ intersect at $S$. The tangent in $S$ to $C_1$ intersects $C_2$ in $A$ different from $S$. The tangent in $S$ to $C_2$ intersects $C_1$ in $B$ different from $S$. Another circle $C_3$ goes through $A, B, S$. The tangent in $S$ to $C_3$ intersects $C_1$ in $P$ different from $S$ and $C_2$ in $Q$ different from $S$. Prove that the distance $PS$ is equal to the distance $QS$.

2024 Baltic Way, 13

Tags: geometry , geometric transformation , reflection , geometry proposed

Let $ABC$ be an acute triangle with orthocentre $H$. Let $D$ be a point outside the circumcircle of triangle $ABC$ such that $\angle ABD=\angle DCA$. The reflection of $AB$ in $BD$ intersects $CD$ at $X$. The reflection of $AC$ in $CD$ intersects $BD$ at $Y$. The lines through $X$ and $Y$ perpendicular to $AC$ and $AB$, respectively, intersect at $P$. Prove that points $D$, $P$ and $H$ are collinear.

2003 Turkey Team Selection Test, 5

Tags: geometry , geometric transformation , reflection , parallelogram , geometry proposed

Let $A$ be a point on a circle with center $O$ and $B$ be the midpoint of $[OA]$. Let $C$ and $D$ be points on the circle such that they lie on the same side of the line $OA$ and $\widehat{CBO} = \widehat{DBA}$. Show that the reflection of the midpoint of $[CD]$ over $B$ lies on the circle.

2010 Contests, 2

Tags: geometry , circumcircle , projective geometry , geometry proposed

Let $ABC$ be a triangle and $L$, $M$, $N$ be the midpoints of $BC$, $CA$ and $AB$, respectively. The tangent to the circumcircle of $ABC$ at $A$ intersects $LM$ and $LN$ at $P$ and $Q$, respectively. Show that $CP$ is parallel to $BQ$.

2007 Romania Team Selection Test, 2

Tags: geometry , geometric transformation , projective geometry , geometry proposed , Desargues , Monge theorem

Let $ABC$ be a triangle, and $\omega_{a}$, $\omega_{b}$, $\omega_{c}$ be circles inside $ABC$, that are tangent (externally) one to each other, such that $\omega_{a}$ is tangent to $AB$ and $AC$, $\omega_{b}$ is tangent to $BA$ and $BC$, and $\omega_{c}$ is tangent to $CA$ and $CB$. Let $D$ be the common point of $\omega_{b}$ and $\omega_{c}$, $E$ the common point of $\omega_{c}$ and $\omega_{a}$, and $F$ the common point of $\omega_{a}$ and $\omega_{b}$. Show that the lines $AD$, $BE$ and $CF$ have a common point.

2010 Baltic Way, 11

Tags: geometry proposed , geometry

Let $ABCD$ be a square and let $S$ be the point of intersection of its diagonals $AC$ and $BD$. Two circles $k,k'$ go through $A,C$ and $B,D$; respectively. Furthermore, $k$ and $k'$ intersect in exactly two different points $P$ and $Q$. Prove that $S$ lies on $PQ$.

2003 CentroAmerican, 4

Tags: geometry , perimeter , parallelogram , geometry proposed

$S_1$ and $S_2$ are two circles that intersect at two different points $P$ and $Q$. Let $\ell_1$ and $\ell_2$ be two parallel lines such that $\ell_1$ passes through the point $P$ and intersects $S_1,S_2$ at $A_1,A_2$ respectively (both distinct from $P$), and $\ell_2$ passes through the point $Q$ and intersects $S_1,S_2$ at $B_1,B_2$ respectively (both distinct from $Q$). Show that the triangles $A_1QA_2$ and $B_1PB_2$ have the same perimeter.

2000 JBMO ShortLists, 22

Tags: geometry , geometry proposed

Consider a quadrilateral with $\angle DAB=60^{\circ}$, $\angle ABC=90^{\circ}$ and $\angle BCD=120^{\circ}$. The diagonals $AC$ and $BD$ intersect at $M$. If $MB=1$ and $MD=2$, find the area of the quadrilateral $ABCD$.

1996 All-Russian Olympiad, 2

Tags: geometry proposed , geometry

The centers $O_1$; $O_2$; $O_3$ of three nonintersecting circles of equal radius are positioned at the vertices of a triangle. From each of the points $O_1$; $O_2$; $O_3$ one draws tangents to the other two given circles. It is known that the intersection of these tangents form a convex hexagon. The sides of the hexagon are alternately colored red and blue. Prove that the sum of the lengths of the red sides equals the sum of the lengths of the blue sides. [i]D. Tereshin[/i]

2024 Baltic Way, 15

Tags: geometry , geometry proposed , circumcircle , combinatorial geometry

There is a set of $N\geq 3$ points in the plane, such that no three of them are collinear. Three points $A$, $B$, $C$ in the set are said to form a [i]Baltic triangle[/i] if no other point in the set lies on the circumcircle of triangle $ABC$. Assume that there exists at least one Baltic triangle. Show that there exist at least $\displaystyle\frac{N}{3}$ Baltic triangles.

2002 Hungary-Israel Binational, 2

Tags: trigonometry , geometry , incenter , inequalities , geometry proposed

Let $A', B' , C'$ be the projections of a point $M$ inside a triangle $ABC$ onto the sides $BC, CA, AB$, respectively. Deﬁne $p(M ) = \frac{MA'\cdot MB'\cdot MC'}{MA \cdot MB \cdot MC}$ . Find the position of point $M$ that maximizes $p(M )$.

2015 Romanian Master of Mathematics, 4

Tags: geometry , geometry proposed

Let $ABC$ be a triangle, and let $D$ be the point where the incircle meets side $BC$. Let $J_b$ and $J_c$ be the incentres of the triangles $ABD$ and $ACD$, respectively. Prove that the circumcentre of the triangle $AJ_bJ_c$ lies on the angle bisector of $\angle BAC$.

2006 Singapore MO Open, 1

Tags: geometry proposed , geometry

In the triangle $ABC,\angle A=\frac{\pi}{3},D,M$ are points on the line $AC$ and $E,N$ are points on the line $AB$ such that $DN$ and $EM$ are the perpendicular bisectors of $AC$ and $AB$ respectively. Let $L$ be the midpoint of $MN$. Prove that $\angle EDL=\angle ELD$

1997 Iran MO (3rd Round), 2

Tags: geometry , parallelogram , geometry proposed

Let $ABC$ and $XYZ$ be two triangles. Define \[A_1=BC\cap ZX, A_2=BC\cap XY,\]\[B_1=CA\cap XY, B_2=CA\cap YZ,\]\[C_1=AB\cap YZ, C_2=AB\cap ZX.\] Hereby, the abbreviation $g\cap h$ means the point of intersection of two lines $g$ and $h$. Prove that $\frac{C_1C_2}{AB}=\frac{A_1A_2}{BC}=\frac{B_1B_2}{CA}$ holds if and only if $\frac{A_1C_2}{XZ}=\frac{C_1B_2}{ZY}=\frac{B_1A_2}{YX}$.

2012 Indonesia TST, 3

Tags: geometry , geometry proposed

Given a convex quadrilateral $ABCD$, let $P$ and $Q$ be points on $BC$ and $CD$ respectively such that $\angle BAP = \angle DAQ$. Prove that the triangles $ABP$ and $ADQ$ have the same area if the line connecting their orthocenters is perpendicular to $AC$.

2005 India IMO Training Camp, 1

Tags: Euler , geometry , circumcircle , incenter , geometry proposed

Let $ABC$ be a triangle with all angles $\leq 120^{\circ}$. Let $F$ be the Fermat point of triangle $ABC$, that is, the interior point of $ABC$ such that $\angle AFB = \angle BFC = \angle CFA = 120^\circ$. For each one of the three triangles $BFC$, $CFA$ and $AFB$, draw its Euler line - that is, the line connecting its circumcenter and its centroid. Prove that these three Euler lines pass through one common point. [i]Remark.[/i] The Fermat point $F$ is also known as the [b]first Fermat point[/b] or the [b]first Toricelli point[/b] of triangle $ABC$. [i]Floor van Lamoen[/i]

2003 Romania Team Selection Test, 14

Tags: geometry , rhombus , geometry proposed

Given is a rhombus $ABCD$ of side 1. On the sides $BC$ and $CD$ we are given the points $M$ and $N$ respectively, such that $MC+CN+MN=2$ and $2\angle MAN = \angle BAD$. Find the measures of the angles of the rhombus. [i]Cristinel Mortici[/i]

2006 Switzerland Team Selection Test, 2

Tags: geometry , circumcircle , power of a point , radical axis , geometry proposed

Let $D$ be inside $\triangle ABC$ and $E$ on $AD$ different to $D$. Let $\omega_1$ and $\omega_2$ be the circumscribed circles of $\triangle BDE$ and $\triangle CDE$ respectively. $\omega_1$ and $\omega_2$ intersect $BC$ in the interior points $F$ and $G$ respectively. Let $X$ be the intersection between $DG$ and $AB$ and $Y$ the intersection between $DF$ and $AC$. Show that $XY$ is $\|$ to $BC$.

2004 Bulgaria National Olympiad, 1

Tags: geometry , incenter , circumcircle , geometry proposed

Let $ I$ be the incenter of triangle $ ABC$, and let $ A_1$, $ B_1$, $ C_1$ be arbitrary points on the segments $ (AI)$, $ (BI)$, $ (CI)$, respectively. The perpendicular bisectors of $ AA_1$, $ BB_1$, $ CC_1$ intersect each other at $ A_2$, $ B_2$, and $ C_2$. Prove that the circumcenter of the triangle $ A_2B_2C_2$ coincides with the circumcenter of the triangle $ ABC$ if and only if $ I$ is the orthocenter of triangle $ A_1B_1C_1$.

2007 Romania Team Selection Test, 2

Tags: geometry , trigonometry , AMC , USA(J)MO , USAJMO , perpendicular bisector , geometry proposed

Let $ ABC$ be a triangle, let $ E, F$ be the tangency points of the incircle $ \Gamma(I)$ to the sides $ AC$, respectively $ AB$, and let $ M$ be the midpoint of the side $ BC$. Let $ N \equal{} AM \cap EF$, let $ \gamma(M)$ be the circle of diameter $ BC$, and let $ X, Y$ be the other (than $ B, C$) intersection points of $ BI$, respectively $ CI$, with $ \gamma$. Prove that \[ \frac {NX} {NY} \equal{} \frac {AC} {AB}. \] [i]Cosmin Pohoata[/i]

2013 JBMO TST - Macedonia, 2

Tags: geometry , geometry proposed

A triangle $ ABC $ is given, and a segment $ PQ=t $ on $ BC $ such that $ P $ is between $ B $ and $ Q $ and $ Q $ is between $ P $ and $ C $. Let $ PP_1 || AB $, $ P_1 $ is on $ AC $, and $ PP_2 || AC $, $ P_2 $ is on $ AB $. Points $ Q_1 $ and $ Q_2 $ аrе defined similar. Prove that the sum of the areas of $ PQQ_1P_1 $ and $ PQQ_2P_2 $ does not depend from the position of $ PQ $ on $ BC $.

2010 Contests, 3

Tags: trigonometry , geometry , angle bisector , geometry proposed

Let $ABC$ be an isosceles triangle with apex at $C.$ Let $D$ and $E$ be two points on the sides $AC$ and $BC$ such that the angle bisectors $\angle DEB$ and $\angle ADE$ meet at $F,$ which lies on segment $AB.$ Prove that $F$ is the midpoint of $AB.$

1994 Vietnam National Olympiad, 2

Tags: geometry , geometric transformation , reflection , trigonometry , circumcircle , geometry proposed

$ABC$ is a triangle. Reflect each vertex in the opposite side to get the triangle $A'B'C'$. Find a necessary and sufficient condition on $ABC$ for $A'B'C'$ to be equilateral.

1992 Turkey Team Selection Test, 2

Tags: geometry , circumcircle , geometry proposed

The line passing through $B$ is perpendicular to the side $AC$ at $E$. This line meets the circumcircle of $\triangle ABC$ at $D$. The foot of the perpendicular from $D$ to the side $BC$ is $F$. If $O$ is the center of the circumcircle of $\triangle ABC$, prove that $BO$ is perpendicular to $EF$.