Olympiad problems

Indonesia MO Shortlist - geometry, g3

Given $ABC$ triangle with incircle $L_1$ and circumcircle $L_2$. If points $X, Y, Z$ lie on $L_2$, such that $XY, XZ$ are tangent to $L_1$, then prove that $YZ$ is also tangent to $L_1$.

2005 Czech And Slovak Olympiad III A, 3

Tags: geometry , tangent , trapezoid

In a trapezoid $ABCD$ with $AB // CD, E$ is the midpoint of $BC$. Prove that if the quadrilaterals $ABED$ and $AECD$ are tangent, then the sides $a = AB, b = BC, c =CD, d = DA$ of the trapezoid satisfy the equalities $a+c = \frac{b}{3} +d$ and $\frac1a +\frac1c = \frac3b$ .

2010 Contests, 3

Tags: combinatorial geometry , combinatorics , hexagon , tangent

On a circular billiard table a ball rebounds from the rails as if the rail was the tangent to the circle at the point of impact. A regular hexagon with its vertices on the circle is drawn on a circular billiard table. A (point-shaped) ball is placed somewhere on the circumference of the hexagon, but not on one of its edges. Describe a periodical track of this ball with exactly four points at the rails. With how many different directions of impact can the ball be brought onto such a track?

2018 Hanoi Open Mathematics Competitions, 12

Tags: geometry , altitude , parallel , circumcircle , tangent

Let $ABC$ be an acute triangle with $AB < AC$, and let $BE$ and $CF$ be the altitudes. Let the median $AM$ intersect $BE$ at point $P$, and let line $CP$ intersect $AB$ at point $D$ (see Figure 2). Prove that $DE \parallel BC$, and $AC$ is tangent to the circumcircle of $\vartriangle DEF$. [img]https://cdn.artofproblemsolving.com/attachments/f/7/bbad9f6019a77c6aa46c3a821857f06233cb93.png[/img]

2012 India Regional Mathematical Olympiad, 1

Tags: radius , geometry , arc , circles , tangent

Let $ABCD$ be a unit square. Draw a quadrant of the a circle with $A$ as centre and $B,D$ as end points of the arc. Similarly, draw a quadrant of a circle with $B$ as centre and $A,C$ as end points of the arc. Inscribe a circle $\Gamma$ touching the arc $AC$ externally, the arc $BD$ externally and also touching the side $AD$. Find the radius of $\Gamma$.

2023 Sharygin Geometry Olympiad, 22

Tags: geometry , tangent , incircle , projective geometry

Let $ABC$ be a scalene triangle, $M$ be the midpoint of $BC,P$ be the common point of $AM$ and the incircle of $ABC$ closest to $A$, and $Q$ be the common point of the ray $AM$ and the excircle farthest from $A$. The tangent to the incircle at $P$ meets $BC$ at point $X$, and the tangent to the excircle at $Q$ meets $BC$ at $Y$. Prove that $MX=MY$.

2023 Argentina National Olympiad, 3

Tags: geometry , tangent

Let $ABC$ be a triangle and $M$ be the middle point of $BC$. Let $\Omega$ be the circumference such as $A,B,C \in \Omega$. Let $P$ be the intersection of $\Omega$ and $AM$. $AF$ is a hight of the triangle, with $F\in BC$, and $H$ the orthocenter.Additionally the intersections of $MH$ and $PF$ with $\Omega$ are $K$ and $T$ respectibly. Demonstrate that the circumscribed circumference of the traingle $KTF$ is tangent with $BC$.

2010 Brazil Team Selection Test, 3

Tags: geometry , power of a point , projective geometry , tangent , cyclic quadrilateral

Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$. [i]Proposed by David Monk, United Kingdom[/i]

Croatia MO (HMO) - geometry, 2014.7

Tags: incenter , circumcircle , geometry , tangent

Let point $I$ be the center of the inscribed circle of an acute-angled triangle $ABC$. Rays $AI$ and $BI$ intersect the circumcircle $k$ of triangle $ABC$ at points $D$ and $E$ respectively. The segments $DE$ and $CA$ intersect at point $F$, the line through point $E$ parallel to the line $FI$ intersects the circle $k$ at point $G$, and the lines $FI$ and $DG$ intersect at point $H$. Prove that the lines $CA$ and $BH$ touch the circumcircle of the triangle $DFH$ at the points $F$ and $H$ respectively.

Russian TST 2021, P3

Tags: geometry , circumcircle , incenter , tangent

Let $ABC$ be a triangle with incenter $I$ and circumcircle $\Gamma$. Circles $\omega_{B}$ passing through $B$ and $\omega_{C}$ passing through $C$ are tangent at $I$. Let $\omega_{B}$ meet minor arc $AB$ of $\Gamma$ at $P$ and $AB$ at $M\neq B$, and let $\omega_{C}$ meet minor arc $AC$ of $\Gamma$ at $Q$ and $AC$ at $N\neq C$. Rays $PM$ and $QN$ meet at $X$. Let $Y$ be a point such that $YB$ is tangent to $\omega_{B}$ and $YC$ is tangent to $\omega_{C}$. Show that $A,X,Y$ are collinear.

2021 Macedonian Mathematical Olympiad, Problem 3

Tags: circles , geometry , trapezoid , parallel lines , angle chasing , tangent , cyclic quadrilateral

Let $ABCD$ be a trapezoid with $AD \parallel BC$ and $\angle BCD < \angle ABC < 90^\circ$. Let $E$ be the intersection point of the diagonals $AC$ and $BD$. The circumcircle $\omega$ of $\triangle BEC$ intersects the segment $CD$ at $X$. The lines $AX$ and $BC$ intersect at $Y$, while the lines $BX$ and $AD$ intersect at $Z$. Prove that the line $EZ$ is tangent to $\omega$ iff the line $BE$ is tangent to the circumcircle of $\triangle BXY$.

2003 Chile National Olympiad, 6

Tags: tangent , circumcircle , geometry

Consider a triangle $ ABC $. On the line $ AC $ take a point $ B_1 $ such that $ AB = AB_1 $ and in addition, $ B_1 $ and $ C $ are located on the same side of the line with respect to the point $ A $. The bisector of the angle $ A $ intersects the side $ BC $ at a point that we will denote as $ A_1 $. Let $ P $ and $ R $ be the circumscribed circles of the triangles $ ABC $ and $ A_1B_1C $ respectively. They intersect at points $ C $ and $ Q $. Prove that the tangent to the circle $ R $ at the point $ Q $ is parallel to the line $ AC $.

2024 Sharygin Geometry Olympiad, 23

Tags: geometry , tangent , transformation

A point $P$ moves along a circle $\Omega$. Let $A$ and $B$ be two fixed points of $\Omega$, and $C$ be an arbitrary point inside $\Omega$. The common external tangents to the circumcircles of triangles $APC$ and $BCP$ meet at point $Q$. Prove that all points $Q$ lie on two fixed lines.

2018 Azerbaijan Junior NMO, 4

Tags: geometry , angle bisector , circles , tangent

A circle $\omega$ and a point $T$ outside the circle is given. Let a tangent from $T$ to $\omega$ touch $\omega$ at $A$, and take points $B,C$ lying on $\omega$ such that $T,B,C$ are colinear. The bisector of $\angle ATC$ intersects $AB$ and $AC$ at $P$ and $Q$,respectively. Prove that $PA=\sqrt{PB\cdot QC}$

1995 China Team Selection Test, 2

Tags: locus problems , tangent , geometry , geometric transformation

Given a fixed acute angle $\theta$ and a pair of internally tangent circles, let the line $l$ which passes through the point of tangency, $A$, cut the larger circle again at $B$ ($l$ does not pass through the centers of the circles). Let $M$ be a point on the major arc $AB$ of the larger circle, $N$ the point where $AM$ intersects the smaller circle, and $P$ the point on ray $MB$ such that $\angle MPN = \theta$. Find the locus of $P$ as $M$ moves on major arc $AB$ of the larger circle.

2012 Polish MO Finals, 5

Tags: geometry , triangle , circumcircle , tangent

Point $O$ is a center of circumcircle of acute triangle $ABC$, bisector of angle $BAC$ cuts side $BC$ in point $D$. Let $M$ be a point such that, $MC \perp BC$ and $MA \perp AD$. Lines $BM$ and $OA$ intersect in point $P$. Show that circle of center in point $P$ passing through a point $A$ is tangent to line $BC$.

1955 Moscow Mathematical Olympiad, 308

Tags: bisect arc , circles , tangent circle , tangent , geometry

* Two circles are tangent to each other externally, and to a third one from the inside. Two common tangents to the first two circles are drawn, one outer and one inner. Prove that the inner tangent divides in halves the arc intercepted by the outer tangent on the third circle.

2002 Junior Balkan Team Selection Tests - Romania, 3

Tags: geometry , tangent , circles , concurrency

Let $C_1(O_1)$ and $ C_2(O_2)$ be two circles such that $C_1$ passes through $O_2$. Point $M$ lies on $C_1$ such that $M \notin O_1O_2$. The tangents from $M$ at $O_2$ meet again $C_1$ at $A$ and $B$. Prove that the tangents from $A$ and $B$ at $C_2$ - others than $MA$ and $MB$ - meet at a point located on $C_1$.

1988 All Soviet Union Mathematical Olympiad, 476

Tags: geometry , circumcircle , tangent , angle bisector

$ABC$ is an acute-angled triangle. The tangents to the circumcircle at $A$ and $C$ meet the tangent at $B$ at $M$ and $N$. The altitude from $B$ meets $AC$ at $P$. Show that $BP$ bisects the angle $MPN$

Kyiv City MO Seniors 2003+ geometry, 2018.11.4

Tags: tangent , geometry , incenter , isosceles

Given an isosceles $ABC$, which has $2AC = AB + BC$. Denote $I$ the center of the inscribed circle, $K$ the midpoint of the arc $ABC$ of the circumscribed circle. Let $T$ be such a point on the line $AC$ that $\angle TIB = 90 {} ^ \circ$. Prove that the line $TB$ touches the circumscribed circle $\Delta KBI$. (Anton Trygub)

1986 ITAMO, 1

Tags: tangent , parallel , circles , geometry

Two circles $\alpha$ and $\beta$ intersect at points $P$ and $Q$. The lines connecting a point $R$ on $\beta$ with $P$ and $Q$ intersect $\alpha$ again at $S$ and $T$ respectively. Prove that $ST$ is parallel to the line tangent to $\beta$ at $R$.

2010 Germany Team Selection Test, 2

Tags: power of a point , tangent , projective geometry , geometry , cyclic quadrilateral

Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$. [i]Proposed by David Monk, United Kingdom[/i]

2024 Kosovo Team Selection Test, P2

Tags: geometry , concurrent lines , beautiful geometry , circles , tangent

Let $\omega$ be a circle and let $A$ be a point lying outside of $\omega$. The tangents from $A$ to $\omega$ touch $\omega$ at points $B$ and $C$. Let $M$ be the midpoint of $BC$ and let $D$ a point on the side $BC$ different from $M$. The circle with diameter $AD$ intersects $\omega$ at points $X$ and $Y$ and the circumcircle of $\bigtriangleup ABC$ again at $E$. Prove that $AD$, $EM$, and $XY$ are concurrent.

2005 Oral Moscow Geometry Olympiad, 6

Tags: tangent , geometry , combinatorial geometry

Six straight lines are drawn on the plane. It is known that for any three of them there is a fourth of the same set of lines, such that all four will touch some circle. Do all six lines necessarily touch the same circle? (I. Bogdanov)

2024 Israel TST, P1

Tags: angle bisector , tangent , geometry

Let $ABC$ be a triangle and let $D$ be a point on $BC$ so that $AD$ bisects the angle $\angle BAC$. The common tangents of the circles $(BAD)$, $(CAD)$ meet at the point $A'$. The points $B'$, $C'$ are defined similarly. Show that $A'$, $B'$, $C'$ are collinear.