Olympiad problems

Mathley 2014-15, 4

Let $(O)$ be the circumcircle of triangle $ABC$, and $P$ a point on the arc $BC$ not containing $A$. $(Q)$ is the $A$-mixtilinear circle of triangle $ABC$, and $(K), (L)$ are the $P$-mixtilinear circles of triangle $PAB, PAC$ respectively. Prove that there is a line tangent to all the three circles $(Q), (K)$ and $(L)$. Nguyen Van Linh, a student at Hanoi Foreign Trade University Cabinet

1989 All Soviet Union Mathematical Olympiad, 489

Tags: geometry , tangent , incircle , common tangent

The incircle of $ABC$ touches $AB$ at $M$. $N$ is any point on the segment $BC$. Show that the incircles of $AMN, BMN, ACN$ have a common tangent.

2008 Oral Moscow Geometry Olympiad, 2

Tags: area , common tangent , geometry

The radii $r$ and $R$ of two non-intersecting circles are given. The common internal tangents of these circles are perpendicular. Find the area of the triangle bounded by these tangents, as well as the common external tangents.

2023 Canadian Mathematical Olympiad Qualification, 3

Tags: common tangent , geometry

Let circles $\Gamma_1$ and $\Gamma_2$ have radii $r_1$ and $r_2$, respectively. Assume that $r_1 < r_2$. Let $T$ be an intersection point of $\Gamma_1$ and $\Gamma_2$, and let $S$ be the intersection of the common external tangents of $\Gamma_1$ and $\Gamma_2$. If it is given that the tangents to $\Gamma_1$ and $ \Gamma_2$ at $T$ are perpendicular, determine the length of $ST$ in terms of $r_1$ and $r_2$.

2011 Sharygin Geometry Olympiad, 7

Tags: circles , equal segments , common tangent , angle , geometry

Circles $\omega$ and $\Omega$ are inscribed into the same angle. Line $\ell$ meets the sides of angles, $\omega$ and $\Omega$ in points $A$ and $F, B$ and $C, D$ and $E$ respectively (the order of points on the line is $A,B,C,D,E, F$). It is known that$ BC = DE$. Prove that $AB = EF$.

1995 Belarus Team Selection Test, 2

Tags: common tangent , geometry , tangent circle , circles

Circles $S,S_1,S_2$ are given in a plane. $S_1$ and $S_2$ touch each other externally, and both touch $S$ internally at $A_1$ and $A_2$ respectively. The common internal tangent to $S_1$ and $S_2$ meets $S$ at $P$ and $Q.$ Let $B_1$ and $B_2$ be the intersections of $PA_1$ and $PA_2$ with $S_1$ and $S_2$, respectively. Prove that $B_1B_2$ is a common tangent to $S_1,S_2$

2021 Israel TST, 3

Tags: orthocenter , common tangent , incircle , geometry

Let $ABC$ be an acute triangle with orthocenter $H$. Prove that there is a line $l$ which is parallel to $BC$ and tangent to the incircles of $ABH$ and $ACH$.

2016 PAMO, 1

Tags: common tangent , geometry

Two circles $\mathcal{C}_1$ and $\mathcal{C}_2$ intersect each other at two distinct points $M$ and $N$. A common tangent lines touches $\mathcal{C}_1$ at $P$ and $\mathcal{C}_2$ at $Q$, the line being closer to $N$ than to $M$. The line $PN$ meets the circle $\mathcal{C}_2$ again at the point $R$. Prove that the line $MQ$ is a bisector of the angle $\angle{PMR}$.

2021 Israel TST, 3

Tags: orthocenter , common tangent , incircle , geometry

Let $ABC$ be an acute triangle with orthocenter $H$. Prove that there is a line $l$ which is parallel to $BC$ and tangent to the incircles of $ABH$ and $ACH$.

2004 Tournament Of Towns, 5

Tags: parabola , common tangent , circles , tangent , geometry

The parabola $y = x^2$ intersects a circle at exactly two points $A$ and $B$. If their tangents at $A$ coincide, must their tangents at $B$ also coincide?

2016 Abels Math Contest (Norwegian MO) Final, 3a

Tags: circles , common tangent , collinear , geometry

Three circles $S_A, S_B$, and $S_C$ in the plane with centers in $A, B$, and $C$, respectively, are mutually tangential on the outside. The touchpoint between $S_A$ and $S_B$ we call $C'$, the one $S_A$ between $S_C$ we call $B'$, and the one between $S_B$ and $S_C$ we call $A'$. The common tangent between $S_A$ and $S_C$ (passing through B') we call $\ell_B$, and the common tangent between $S_B$ and $S_C$ (passing through $A'$) we call $\ell_A$. The intersection point of $\ell_A$ and $\ell_B$ is called $X$. The point $Y$ is located so that $\angle XBY$ and $\angle YAX$ are both right angles. Show that the points $X, Y$, and $C'$ lie on a line if and only if $AC = BC$.

Indonesia MO Shortlist - geometry, g2

Tags: geometry , concyclic , common tangent

It is known that two circles have centers at $P$ and $Q$. Prove that the intersection points of the two internal common tangents of the two circles with their two external common tangents lie on the same circle.

2012 Peru MO (ONEM), 4

Tags: right angle , mixtilinear , common tangent , circles , geometry

In a circle $S$, a chord $AB$ is drawn and let $M$ be the midpoint of the arc $AB$. Let $P$ be a point in segment $AB$ other than its midpoint. The extension of the segment $MP$ cuts $S$ in $Q$. Let $S_1$ be the circle that is tangent to the AP segments and $MP$, and also is tangent to $S$, and let $S_2$ be the circle that is tangent to the segments $BP$ and $MP$, and also tangent to $S$. The common outer tangent lines to the circles $S_1$ and $S_2$ are cut at $C$. Prove that $\angle MQC = 90^o$.

1963 Czech and Slovak Olympiad III A, 3

Tags: locus , common tangent , geometry , tangent circle

A line $MN$ is given in the plane. Consider circles $k_1$, $k_2$ tangent to the line at points $M$, $N$, respectively, while touching each other externally. Let $X$ be the midpoint of the segment $PQ$, where $P$, $Q$ are in this order tangent points of the second common external tangent of the circles $k_1$, $k_2$. Find the locus of the points $X$ for all pairs of circles of the specified properties.

1988 Mexico National Olympiad, 3

Tags: common tangent , tangent circle , area of a triangle , geometry

Two externally tangent circles with different radii are given. Their common tangents form a triangle. Find the area of this triangle in terms of the radii of the two circles.

2014 Sharygin Geometry Olympiad, 4

Tags: circles , geometry , common tangent

Let $H$ be the orthocenter of a triangle $ABC$. Given that $H$ lies on the incircle of $ABC$ , prove that three circles with centers $A, B, C$ and radii $AH, BH, CH$ have a common tangent. (Mahdi Etesami Fard)

2019 Tuymaada Olympiad, 2

Tags: circumcircle , common tangent , circles , geometry

A triangle $ABC$ with $AB < AC$ is inscribed in a circle $\omega$. Circles $\gamma_1$ and $\gamma_2$ touch the lines $AB$ and $AC$, and their centres lie on the circumference of $\omega$. Prove that $C$ lies on a common external tangent to $\gamma_1$ and $\gamma_2$.

2022 Dutch IMO TST, 2

Tags: geometry , common tangent , concurrency

Two circles $\Gamma_1$ and $\Gamma_2$are given with centres $O_1$ and $O_2$ and common exterior tangents $\ell_1$ and $\ell_2$. The line $\ell_1$ intersects $\Gamma_1$ in $A$ and $\Gamma_2$ in $B$. Let $X$ be a point on segment $O_1O_2$, not lying on $\Gamma_1$ or $\Gamma_2$. The segment $AX$ intersects $\Gamma_1$ in $Y \ne A$ and the segment $BX$ intersects $\Gamma_2$ in $Z \ne B$. Prove that the line through $Y$ tangent to $\Gamma_1$ and the line through $Z$ tangent to $\Gamma_2$ intersect each other on $\ell_2$.

2010 Sharygin Geometry Olympiad, 4

Tags: circles , common tangent , equal angles , geometry , tangent

Circles $\omega_1$ and $\omega_2$ inscribed into equal angles $X_1OY$ and $Y OX_2$ touch lines $OX_1$ and $OX_2$ at points $A_1$ and $A_2$ respectively. Also they touch $OY$ at points $B_1$ and $B_2$. Let $C_1$ be the second common point of $A_1B_2$ and $\omega_1, C_2$ be the second common point of $A_2B_1$ and $\omega_2$. Prove that $C_1C_2$ is the common tangent of two circles.

1997 Slovenia Team Selection Test, 1

Tags: common tangent , tangent circle , tangent , geometry

Circles $K_1$ and $K_2$ are externally tangent to each other at $A$ and are internally tangent to a circle $K$ at $A_1$ and $A_2$ respectively. The common tangent to $K_1$ and $K_2$ at $A$ meets $K$ at point $P$. Line $PA_1$ meets $K_1$ again at $B_1$ and $PA_2$ meets $K_2$ again at $B_2$. Show that $B_1B_2$ is a common tangent of $K_1$ and $K_2$.

Swiss NMO - geometry, 2009.7

Tags: concurrency , circles , geometry , common tangent

Points $A, M_1, M_2$ and $C$ are on a line in this order. Let $k_1$ the circle with center $M_1$ passing through $A$ and $k_2$ the circle with center $M_2$ passing through $C$. The two circles intersect at points $E$ and $F$. A common tangent of $k_1$ and $k_2$, touches $k_1$ at $B$ and $k_2$ at $D$. Show that the lines $AB, CD$ and $EF$ intersect at one point.

1969 Czech and Slovak Olympiad III A, 6

Tags: 3d geometry , common tangent , tangent circle , sphere , geometry

A sphere with unit radius is given. Furthermore, circles $k_0,k_1,\ldots,k_n\ (n\ge3)$ of the same radius $r$ are given on the sphere. The circle $k_0$ is tangent to all other circles $k_i$ and every two circles $k_i,k_{i+1}$ are tangent for $i=1,\ldots,n$ (assuming $k_{n+1}=k_1$). a) Find relation between numbers $n,r.$ b) Determine for which $n$ the described situation can occur and compute the corresponding radius $r.$ (We say non-planar circles are tangent if they have only a single common point and their tangent lines in this point coincide.)

2022 Dutch IMO TST, 2

Tags: concurrency , geometry , common tangent

Two circles $\Gamma_1$ and $\Gamma_2$are given with centres $O_1$ and $O_2$ and common exterior tangents $\ell_1$ and $\ell_2$. The line $\ell_1$ intersects $\Gamma_1$ in $A$ and $\Gamma_2$ in $B$. Let $X$ be a point on segment $O_1O_2$, not lying on $\Gamma_1$ or $\Gamma_2$. The segment $AX$ intersects $\Gamma_1$ in $Y \ne A$ and the segment $BX$ intersects $\Gamma_2$ in $Z \ne B$. Prove that the line through $Y$ tangent to $\Gamma_1$ and the line through $Z$ tangent to $\Gamma_2$ intersect each other on $\ell_2$.

1996 Tournament Of Towns, (489) 2

Tags: common tangent , geometry

An exterior common tangent to two non-intersecting circles with centers and $O_2$ touches them at the points $A_1$ and $A_2$ respectively. The segment $O_1O_2$ intersects the circles at the points $B_1$ and $B_2$ respectively. $C$ is the point where the straight lines $A_1B_1$ and $A_2B_2$ meet. $D$ is the point on the line $A_1A_2$ such that $CD$ is perpendicular to $B_1B_2$. Prove that $A_1D = DA_2$.

2013 Oral Moscow Geometry Olympiad, 2

Tags: angle bisector , geometry , equal angles , inscribed circle , common tangent

Inside the angle $AOD$, the rays $OB$ and $OC$ are drawn such that $\angle AOB = \angle COD.$ Two circles are inscribed inside the angles $\angle AOB$ and $\angle COD$ . Prove that the intersection point of the common internal tangents of these circles lies on the bisector of the angle $AOD$.