Olympiad problems

2014 Taiwan TST Round 3, 1

In convex hexagon $ABCDEF$, $AB \parallel DE$, $BC \parallel EF$, $CD \parallel FA$, and \[ AB+DE = BC+EF = CD+FA. \] The midpoints of sides $AB$, $BC$, $DE$, $EF$ are $A_1$, $B_1$, $D_1$, $E_1$, and segments $A_1D_1$ and $B_1E_1$ meet at $O$. Prove that $\angle D_1OE_1 = \frac12 \angle DEF$.

2014 Contests, 2

Tags: geometry , geometry proposed

Let $ABCD$ be a square. Let $N,P$ be two points on sides $AB, AD$, respectively such that $NP=NC$, and let $Q$ be a point on $AN$ such that $\angle QPN = \angle NCB$. Prove that \[ \angle BCQ = \dfrac{1}{2} \angle AQP .\]

2009 Junior Balkan Team Selection Test, 2

Tags: ratio , trigonometry , geometry proposed , geometry

In isosceles right triangle $ ABC$ a circle is inscribed. Let $ CD$ be the hypotenuse height ($ D\in AB$), and let $ P$ be the intersection of inscribed circle and height $ CD$. In which ratio does the circle divide segment $ AP$?

2007 USA Team Selection Test, 1

Tags: Asymptote , geometry , geometric transformation , geometry proposed

Circles $ \omega_1$ and $ \omega_2$ meet at $ P$ and $ Q$. Segments $ AC$ and $ BD$ are chords of $ \omega_1$ and $ \omega_2$ respectively, such that segment $ AB$ and ray $ CD$ meet at $ P$. Ray $ BD$ and segment $ AC$ meet at $ X$. Point $ Y$ lies on $ \omega_1$ such that $ PY \parallel BD$. Point $ Z$ lies on $ \omega_2$ such that $ PZ \parallel AC$. Prove that points $ Q,X,Y,Z$ are collinear.

2000 JBMO ShortLists, 18

Tags: geometry , geometry proposed

A triangle $ABC$ is given. Find all the segments $XY$ that lie inside the triangle such that $XY$ and five of the segments $XA,XB, XC, YA,YB,YC$ divide the triangle $ABC$ into $5$ regions with equal areas. Furthermore, prove that all the segments $XY$ have a common point.

2008 Sharygin Geometry Olympiad, 11

Tags: geometry proposed , geometry

(A.Zaslavsky, 9--10) Given four points $ A$, $ B$, $ C$, $ D$. Any two circles such that one of them contains $ A$ and $ B$, and the other one contains $ C$ and $ D$, meet. Prove that common chords of all these pairs of circles pass through a fixed point.

2007 Iran Team Selection Test, 1

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

In an isosceles right-angled triangle shaped billiards table , a ball starts moving from one of the vertices adjacent to hypotenuse. When it reaches to one side then it will reflect its path. Prove that if we reach to a vertex then it is not the vertex at initial position [i]By Sam Nariman[/i]

2013 Vietnam National Olympiad, 3

Tags: geometry , geometric transformation , reflection , perpendicular bisector , geometry proposed

Let $ABC$ be a triangle such that $ABC$ isn't a isosceles triangle. $(I)$ is incircle of triangle touches $BC,CA,AB$ at $D,E,F$ respectively. The line through $E$ perpendicular to $BI$ cuts $(I)$ again at $K$. The line through $F$ perpendicular to $CI$ cuts $(I)$ again at $L$.$J$ is midpoint of $KL$. [b]a)[/b] Prove that $D,I,J$ collinear. [b]b)[/b] $B,C$ are fixed points,$A$ is moved point such that $\frac{AB}{AC}=k$ with $k$ is constant.$IE,IF$ cut $(I)$ again at $M,N$ respectively.$MN$ cuts $IB,IC$ at $P,Q$ respectively. Prove that bisector perpendicular of $PQ$ through a fixed point.

2005 Nordic, 4

Tags: geometry , geometric transformation , homothety , LaTeX , geometry proposed

The circle $\zeta_{1}$ is inside the circle $\zeta_{2}$, and the circles touch each other at $A$. A line through $A$ intersects $\zeta_{1}$ also at $B$, and $\zeta_{2}$ also at $C$. The tangent to $\zeta_{1}$ at $B$ intersects $\zeta_{2}$ at $D$ and $E$. The tangents of $\zeta_{1}$ passing thorugh $C$ touch $\zeta_{2}$ at $F$ and $G$. Prove that $D$, $E$, $F$ and $G$ are concyclic.

2005 QEDMO 1st, 9 (G3)

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

Let $ABC$ be a triangle with $AB\neq CB$. Let $C^{\prime}$ be a point on the ray $[AB$ such that $AC^{\prime}=CB$. Let $A^{\prime}$ be a point on the ray $[CB$ such that $CA^{\prime}=AB$. Let the circumcircles of triangles $ABA^{\prime}$ and $CBC^{\prime}$ intersect at a point $Q$ (apart from $B$). Prove that the line $BQ$ bisects the segment $CA$. Darij

2014 Dutch IMO TST, 4

Tags: geometry , geometric transformation , reflection , angle bisector , geometry proposed

Let $\triangle ABC$ be a triangle with $|AC|=2|AB|$ and let $O$ be its circumcenter. Let $D$ be the intersection of the bisector of $\angle A$ with $BC$. Let $E$ be the orthogonal projection of $O$ to $AD$ and let $F\ne D$ be the point on $AD$ satisfying $|CD|=|CF|$. Prove that $\angle EBF=\angle ECF$.

1999 Baltic Way, 14

Tags: geometry , ratio , geometry proposed

Let $ABC$ be an isosceles triangle with $AB=AC$. Points $D$ and $E$ lie on the sides $AB$ and $AC$, respectively. The line passing through $B$ and parallel to $AC$ meets the line $DE$ at $F$. The line passing through $C$ and parallel to $AB$ meets the line $DE$ at $G$. Prove that \[\frac{[DBCG]}{[FBCE]}=\frac{AD}{DE} \]

2005 District Olympiad, 3

Tags: geometry , 3D geometry , tetrahedron , circumcircle , geometry proposed

Let $O$ be a point equally distanced from the vertices of the tetrahedron $ABCD$. If the distances from $O$ to the planes $(BCD)$, $(ACD)$, $(ABD)$ and $(ABC)$ are equal, prove that the sum of the distances from a point $M \in \textrm{int}[ABCD]$, to the four planes, is constant.

2003 Romania Team Selection Test, 8

Tags: geometry , circumcircle , invariant , geometry proposed

Two circles $\omega_1$ and $\omega_2$ with radii $r_1$ and $r_2$, $r_2>r_1$, are externally tangent. The line $t_1$ is tangent to the circles $\omega_1$ and $\omega_2$ at points $A$ and $D$ respectively. The parallel line $t_2$ to the line $t_1$ is tangent to the circle $\omega_1$ and intersects the circle $\omega_2$ at points $E$ and $F$. The line $t_3$ passing through $D$ intersects the line $t_2$ and the circle $\omega_2$ in $B$ and $C$ respectively, both different of $E$ and $F$ respectively. Prove that the circumcircle of the triangle $ABC$ is tangent to the line $t_1$. [i]Dinu Serbanescu[/i]

2006 Macedonia National Olympiad, 4

Tags: geometry , circumcircle , trigonometry , complex numbers , trig identities , Law of Sines , geometry proposed

Let $M$ be a point on the smaller arc $A_1A_n$ of the circumcircle of a regular $n$-gon $A_1A_2\ldots A_n$ . $(a)$ If $n$ is even, prove that $\sum_{i=1}^n(-1)^iMA_i^2=0$. $(b)$ If $n$ is odd, prove that $\sum_{i=1}^n(-1)^iMA_i=0$.

2000 Bulgaria National Olympiad, 2

Tags: geometry , geometric transformation , reflection , geometry proposed

Let be given an acute triangle $ABC$. Show that there exist unique points $A_1 \in BC$, $B_1 \in CA$, $C_1 \in AB$ such that each of these three points is the midpoint of the segment whose endpoints are the orthogonal projections of the other two points on the corresponding side. Prove that the triangle $A_1B_1C_1$ is similar to the triangle whose side lengths are the medians of $\triangle ABC$.

2014 District Olympiad, 4

Tags: number theory , least common multiple , geometry proposed , geometry

Let $ABCD$ be a square and consider the points $K\in AB, L\in BC,$ and $M\in CD$ such that $\Delta KLM$ is a right isosceles triangle, with the right angle at $L$. Prove that the lines $AL$ and $DK$ are perpendicular to each other.

2014 Baltic Way, 13

Tags: geometry , trigonometry , circumcircle , trig identities , Law of Sines , geometry proposed

Let $ABCD$ be a square inscribed in a circle $\omega$ and let $P$ be a point on the shorter arc $AB$ of $\omega$. Let $CP\cap BD = R$ and $DP \cap AC = S.$ Show that triangles $ARB$ and $DSR$ have equal areas.

1996 All-Russian Olympiad, 3

Tags: geometry , 3D geometry , pyramid , geometry proposed

Show that for $n\ge 5$, a cross-section of a pyramid whose base is a regular $n$-gon cannot be a regular $(n + 1)$-gon. [i]N. Agakhanov, N. Tereshin[/i]

2020 Turkey EGMO TST, 5

Tags: geometry , geometry proposed

$A, B, C, D, E$ points are on $\Gamma$ cycle clockwise. $[AE \cap [CD = \{M\}$ and $[AB \cap [DC = \{N\}$. The line parallels to $EC$ and passes through $M$ intersects with the line parallels to $BC$ and passes through $N$ on $K$. Similarly, the line parallels to $ED$ and passes through $M$ intersects with the line parallels to $BD$ and passes through $N$ on $L$. Show that the lines $LD$ and $KC$ intersect on $\Gamma$.

2009 Sharygin Geometry Olympiad, 24

Tags: geometry , 3D geometry , sphere , pyramid , geometry proposed

A sphere is inscribed into a quadrangular pyramid. The point of contact of the sphere with the base of the pyramid is projected to the edges of the base. Prove that these projections are concyclic.

1986 IMO Longlists, 3

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

A line parallel to the side $BC$ of a triangle $ABC$ meets $AB$ in $F$ and $AC$ in $E$. Prove that the circles on $BE$ and $CF$ as diameters intersect in a point lying on the altitude of the triangle $ABC$ dropped from $A$ to $BC.$

2013 ITAMO, 2

Tags: geometry , geometry proposed

In triangle $ABC$, suppose we have $a> b$, where $a=BC$ and $b=AC$. Let $M$ be the midpoint of $AB$, and $\alpha, \beta$ are inscircles of the triangles $ACM$ and $BCM$ respectively. Let then $A'$ and $B'$ be the points of tangency of $\alpha$ and $\beta$ on $CM$. Prove that $A'B'=\frac{a - b}{2}$.

2009 JBMO TST - Macedonia, 3

Tags: geometry proposed , geometry

Let $ \triangle ABC $ be equilateral. On the side $ AB $ points $ C_{1} $ and $ C_{2} $, on the side $ AC $ points $ B_{1} $ and $ B_{2} $ are chosen, and on the side $ BC $ points $ A_{1} $ and $ A_{2} $ are chosen. The following condition is given : $ A_{1}A_{2} $ = $ B_{1}B_{2} $ = $ C_{1}C_{2} $. Let the intersection lines $ A_{2}B_{1}$ and $ B_{2}C_{1} $, $ B_{2}C_{1} $ and $ C_{2}A_{1} $ and $ C_{2}A_{1} $ and $ A_{2}B_{1} $ are $ E $, $ F $, and $ G $ respectively. Show that the triangle formed by $ B_{1}A_{2} $, $ A_{1}C_{2} $ and $ C_{1}B_{2} $ is similar to $ \triangle EFG $.

1989 Iran MO (2nd round), 3

Tags: geometry , geometric transformation , reflection , geometry proposed

A line $d$ is called [i]faithful[/i] to triangle $ABC$ if $d$ be in plane of triangle $ABC$ and the reflections of $d$ over the sides of $ABC$ be concurrent. Prove that for any two triangles with acute angles lying in the same plane, either there exists exactly one [i]faithful[/i] line to both of them, or there exist infinitely [i]faithful[/i] lines to them.