Olympiad problems

2004 Postal Coaching, 9

Tags: geometry

Let $ABCDEF$ be a regular hexagon of side lengths $1$ and $O$ its centre, Join $O$ cto each of the six vertices , thus getting $12$ unit line segments in all. Find the number of closed paths from (i) $O$ to $O$ (ii) $A$ to $A$ each of length $2004$

2003 Bulgaria Team Selection Test, 5

Tags: law of sines , trig identities , asymptote , geometry , trigonometry

Let $ABCD$ be a circumscribed quadrilateral and let $P$ be the orthogonal projection of its in center on $AC$. Prove that $\angle {APB}=\angle {APD}$

2020 Greece Junior Math Olympiad, 2

Tags: geometry , altitude , angle

Let $ABC$ be an acute-angled triangle with $AB<AC$. Let $D$ be the midpoint of side $BC$ and $BE,CZ$ be the altitudes of the triangle $ABC$. Line $ZE$ intersects line $BC$ at point $O$. (i) Find all the angles of the triangle $ZDE$ in terms of angle $\angle A$ of the triangle $ABC$ (ii) Find the angle $\angle BOZ$ in terms of angles $\angle B$ and $\angle C$ of the triangle $ABC$

2022 Sharygin Geometry Olympiad, 1

Tags: geometry

Let $O$ and $H$ be the circumcenter and the orthocenter respectively of triangle $ABC$. Itis known that $BH$ is the bisector of angle $ABO$. The line passing through $O$ and parallel to $AB$ meets $AC$ at $K$. Prove that $AH = AK$

1995 Tournament Of Towns, (471) 5

Tags: combinatorial geometry , combinatorics , geometry

A simple polygon in the plane is a figure bounded by a closed nonself-intersecting broken line. (a) Do there exist two congruent simple $7$-gons in the plane such that all the seven vertices of one of the $7$-gons are the vertices of the other one and yet these two $7$-gons have no common sides? (b) Do there exist three such $7$-gons? (V Proizvolov)

2022 Adygea Teachers' Geometry Olympiad, 1

Tags: geometry

In triangle $ABC$, $\angle A = 60^o$,$ \angle B = 45^o$. On the sides $AC$ and $BC$ points $M$ and $N$ are taken, respectively, so that the straight line $MN$ cuts off a triangle similar to this one. Find the ratio of $MN$ to $AB$ if it is known that $CM : AM = 2:1$.

1952 Poland - Second Round, 2

Tags: area , geometry

Prove that if $ a $, $ b $, $ c $, $ d $ are the sides of a quadrilateral in which a circle can be circumscribed and a circle can be inscribed in it, then the area $ S $ of the quadrilateral is given by $$S = \sqrt{abcd}.$$

2017 239 Open Mathematical Olympiad, 5

Tags: geometry

Given a quadrilateral $ABCD$ in which$$\sqrt{2}(BC-BA)=AC.$$Let $X$ be the midpoint of $AC$. Prove that $2\angle BXD=\angle DAB - \angle DCB.$

1971 IMO Shortlist, 2

Tags: euclidean distance , point set , combinatorial geometry , geometry

Prove that for every positive integer $m$ we can find a finite set $S$ of points in the plane, such that given any point $A$ of $S$, there are exactly $m$ points in $S$ at unit distance from $A$.

2024 USEMO, 3

Tags: geometry

Let $ABC$ be a triangle with incenter $I$. Two distinct points $P$ and $Q$ are chosen on the circumcircle of $ABC$ such that \[ \angle API = \angle AQI = 45^\circ. \] Lines $PQ$ and $BC$ meet at $S$. Let $H$ denote the foot of the altitude from $A$ to $BC$. Prove that $\angle AHI = \angle ISH$. [i]Matsvei Zorka[/i]

2008 Romania Team Selection Test, 3

Tags: circumcircle , geometry

Let $ ABCDEF$ be a convex hexagon with all the sides of length 1. Prove that one of the radii of the circumcircles of triangles $ ACE$ or $ BDF$ is at least 1.

1967 IMO Longlists, 25

Tags: 3d geometry , sphere , angle , geometry

Three disks of diameter $d$ are touching a sphere in their centers. Besides, every disk touches the other two disks. How to choose the radius $R$ of the sphere in order that axis of the whole figure has an angle of $60^\circ$ with the line connecting the center of the sphere with the point of the disks which is at the largest distance from the axis ? (The axis of the figure is the line having the property that rotation of the figure of $120^\circ$ around that line brings the figure in the initial position. Disks are all on one side of the plane, passing through the center of the sphere and orthogonal to the axis).

2022 JHMT HS, 10

Tags: circumcircle , geometry , incenter

In $\triangle JMT$, $JM=410$, $JT=49$, and $\angle{MJT}>90^\circ$. Let $I$ and $H$ be the incenter and orthocenter of $\triangle JMT$, respectively. The circumcircle of $\triangle JIH$ intersects $\overleftrightarrow{JT}$ at a point $P\neq J$, and $IH=HP$. Find $MT$.

2018 HMNT, 3

Tags: geometry

$HOW,BOW,$ and $DAH$ are equilateral triangles in a plane such that $WO=7$ and $AH=2$. Given that $D,A,B$ are collinear in that order, find the length of $BA$.

2014 Junior Balkan Team Selection Tests - Moldova, 7

Tags: right triangle , equal segments , angle , isosceles , geometry

Let the isosceles right triangle $ABC$ with $\angle A= 90^o$ . The points $E$ and $F$ are taken on the ray $AC$ so that $\angle ABE = 15^o$ and $CE = CF$. Determine the measure of the angle $CBF$.

2000 Turkey Junior National Olympiad, 1

Tags: angle bisector , geometry

Let $ABC$ be a triangle with $\angle BAC = 90^\circ$. Construct the square $BDEC$ such as $A$ and the square are at opposite sides of $BC$. Let the angle bisector of $\angle BAC$ cut the sides $[BC]$ and $[DE]$ at $F$ and $G$, respectively. If $|AB|=24$ and $|AC|=10$, calculate the area of quadrilateral $BDGF$.

2014 Indonesia MO Shortlist, G1

Tags: geometric transformation , reflection , geometry

The inscribed circle of the $ABC$ triangle has center $I$ and touches to $BC$ at $X$. Suppose the $AI$ and $BC$ lines intersect at $L$, and $D$ is the reflection of $L$ wrt $X$. Points $E$ and $F$ respectively are the result of a reflection of $D$ wrt to lines $CI$ and $BI$ respectively. Show that quadrilateral $BCEF$ is cyclic .

2017 Sharygin Geometry Olympiad, 1

Tags: circumcircle , geometry

If two circles intersect at $A,B$ and common tangents of them intesrsect circles at $C,D$if $O_a$is circumcentre of $ACD$ and $O_b$ is circumcentre of $BCD$ prove $AB$ intersects $O_aO_b$ at its midpoint

2022 Malaysia IMONST 2, 1

Tags: cyclic quadrilateral , geometry

Given a circle and a quadrilateral $ABCD$ whose vertices all lie on the circle. Let $R$ be the midpoint of arc $AB$. The line $RC$ meets line $AB$ at point $S$, and the line $RD$ meets line $AB$ at point $T$. Prove that $CDTS$ is a cyclic quadrilateral.

2009 Kazakhstan National Olympiad, 5

Tags: geometry

Quadrilateral $ABCD$ inscribed in circle with center $O$. Let lines $AD$ and $BC$ intersects at $M$, lines $AB$ and $CD$- at $N$, lines $AC$ and $BD$ -at $P$, lines $OP$ and $MN$ at $K$. Proved that $ \angle AKP = \angle PKC$. As I know, this problem was very short solution by polars, but in olympiad for this solution gives maximum 4 balls (in marking schemes written, that needs to prove all theorems about properties of polars)

2009 All-Russian Olympiad, 3

Tags: symmetry , 3d geometry , pyramid , perpendicular bisector , sphere , geometry , circumcircle

Let $ ABCD$ be a triangular pyramid such that no face of the pyramid is a right triangle and the orthocenters of triangles $ ABC$, $ ABD$, and $ ACD$ are collinear. Prove that the center of the sphere circumscribed to the pyramid lies on the plane passing through the midpoints of $ AB$, $ AC$ and $ AD$.

2018 Stanford Mathematics Tournament, 2

Tags: geometry , 3d geometry

What is the largest possible height of a right cylinder with radius $3$ that can fit in a cube with side length $12$?

Croatia MO (HMO) - geometry, 2011.7

Tags: perpendicular bisector , power of a point , radical axis , geometry , projective geometry , circumcircle

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 $KL$.

2012 Irish Math Olympiad, 2

Tags: circumcircle , geometry , angle bisector

Consider a triangle $ABC$ with $|AB|\neq |AC|$. The angle bisector of the angle $CAB$ intersects the circumcircle of $\triangle ABC$ at two points $A$ and $D$. The circle of center $D$ and radius $|DC|$ intersects the line $AC$ at two points $C$ and $B’$. The line $BB’$ intersects the circumcircle of $\triangle ABC$ at $B$ and $E$. Prove that $B’$ is the orthocenter of $\triangle AED$.

2013 ELMO Shortlist, 13

Tags: ratio , perpendicular bisector , angle bisector , circumcircle , asymptote , power of a point , geometry

In $\triangle ABC$, $AB<AC$. $D$ and $P$ are the feet of the internal and external angle bisectors of $\angle BAC$, respectively. $M$ is the midpoint of segment $BC$, and $\omega$ is the circumcircle of $\triangle APD$. Suppose $Q$ is on the minor arc $AD$ of $\omega$ such that $MQ$ is tangent to $\omega$. $QB$ meets $\omega$ again at $R$, and the line through $R$ perpendicular to $BC$ meets $PQ$ at $S$. Prove $SD$ is tangent to the circumcircle of $\triangle QDM$. [i]Proposed by Ray Li[/i]