Olympiad problems

2002 Estonia National Olympiad, 4

A convex quadrilateral $ABCD$ is inscribed in a circle $\omega$. The rays $AD$ and $BC$ meet in point $K$ and the rays $AB$ and $DC$ meet in $L$. Prove that the circumcircle of triangle $AKL$ is tangent to $\omega$ if and only if so is the circumcircle of triangle $CKL$.

2017 Saudi Arabia Pre-TST + Training Tests, 3

Tags: circumcircle , tangent circle , geometry

Let $ABCD$ be a convex quadrilateral. Ray $AD$ meets ray $BC$ at $P$. Let $O,O'$ be the circumcenters of triangles $PCD, PAB$, respectively, $H,H'$ be the orthocenters of triangles $PCD, PAB$, respectively. Prove that circumcircle of triangle $DOC$ is tangent to circumcircle of triangle $AO'B$ if and only if circumcircle of triangle $DHC$ is tangent to circumcircle of triangle $AH'B$.

2021 Saudi Arabia IMO TST, 3

Tags: geometry , excenter , computer problems , tangent circle , incenter

Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other

Indonesia MO Shortlist - geometry, g2

Tags: circles , collinear , perpendicular , tangent circle , geometry

Two circles that are not equal are tangent externally at point $R$. Suppose point $P$ is the intersection of the external common tangents of the two circles. Let $A$ and $B$ are two points on different circles so that $RA$ is perpendicular to $RB$. Show that the line $AB$ passes through $P$.

1989 Mexico National Olympiad, 5

Tags: rectangle , geometry , tangent circle

Let $C_1$ and $C_2$ be two tangent unit circles inside a circle $C$ of radius $2$. Circle $C_3$ inside $C$ is tangent to the circles $C,C_1,C_2$, and circle $C_4$ inside $C$ is tangent to $C,C_1,C_3$. Prove that the centers of $C,C_1,C_3$ and $C_4$ are vertices of a rectangle.

2019 Yasinsky Geometry Olympiad, p3

Tags: tangent circle , circles , angle bisector , geometry

Two circles $\omega_1$ and $\omega_2$ are tangent externally at the point $P$. Through the point $A$ of the circle $\omega_1$ is drawn a tangent to this circle, which intersects the circle $\omega_2$ at points $B$ and $C$ (see figure). Line $CP$ intersects again the circle $\omega_1$ to $D$. Prove that the $PA$ is a bisector of the angle $DPB$. [img]https://1.bp.blogspot.com/-nmKZGdBXfao/XOd51gRFuyI/AAAAAAAAKO0/EYo2SCW0eGcJsF64-Avo6w73ugkIIQ30ACK4BGAYYCw/s1600/Yasinsky%2B2019%2Bp2.png[/img]

Kyiv City MO Seniors 2003+ geometry, 2017.11.5.1

Tags: circumcircle , tangent circle , geometry

The bisector $AD$ is drawn in the triangle $ABC$. Circle $k$ passes through the vertex $A$ and touches the side $BC$ at point $D$. Prove that the circle circumscribed around $ABC$ touches the circle $k$ at point $A$.

1952 Moscow Mathematical Olympiad, 223

Tags: geometry , incircle , tangent circle

In a convex quadrilateral $ABCD$, let $AB + CD = BC + AD$. Prove that the circle inscribed in $ABC$ is tangent to the circle inscribed in $ACD$.

2014 Hanoi Open Mathematics Competitions, 7

Tags: locus , tangent circle , geometry

Let two circles $C_1,C_2$ with different radius be externally tangent at a point $T$. Let $A$ be on $C_1$ and $B$ be on $C_2$, with $A,B \ne T$ such that $\angle ATB = 90^o$. (a) Prove that all such lines $AB$ are concurrent. (b) Find the locus of the midpoints of all such segments $AB$.

2001 239 Open Mathematical Olympiad, 5

Tags: circles , geometry , tangent circle , equal angles

The circles $ S_1 $ and $ S_2 $ intersect at points $ A $ and $ B $. Circle $ S_3 $ externally touches $ S_1 $ and $ S_2 $ at points $ C $ and $ D $ respectively. Let $ K $ be the midpoint of the chord cut by the line $ AB $ on circles $ S_3 $. Prove that $ \angle CKA = \angle DKA $.

Cono Sur Shortlist - geometry, 2003.G2

Tags: diameter , tangent circle , geometry

The circles $C_1, C_2$ and $C_3$ are externally tangent in pairs (each tangent to other two externally). Let $M$ the common point of $C_1$ and $C_2, N$ the common point of $C_2$ and $C_3$ and $P$ the common point of $C_3$ and $C_1$. Let $A$ be an arbitrary point of $C_1$. Line $AM$ cuts $C_2$ in $B$, line $BN$ cuts $C_3$ in $C$ and line $CP$ cuts $C_1$ in $D$. Prove that $AD$ is diameter of $C_1$.

Estonia Open Junior - geometry, 2020.2.5

Tags: geometry , angle , tangent circle

The circle $\omega_2$ passing through the center $O$ of the circle $\omega_1$, is tangent to the circle $\omega_2$ at the point $A$. On the circle $\omega_2$, the point $C$ is taken so that the ray $AC$ intersects the circle $\omega_1$ for second time at point $D$, the ray $OC$ intersects the circle $\omega_1$ at point $E$ and the lines $DE$ and $AO$ are parallel. Find the size of the angle $DAE$.

2021 Mexico National Olympiad, 4

Tags: circumcircle , geometry , euler line , tangent circle , orthocenter

Let $ABC$ be an acutangle scalene triangle with $\angle BAC = 60^{\circ}$ and orthocenter $H$. Let $\omega_b$ be the circumference passing through $H$ and tangent to $AB$ at $B$, and $\omega_c$ the circumference passing through $H$ and tangent to $AC$ at $C$. [list] [*] Prove that $\omega_b$ and $\omega_c$ only have $H$ as common point. [*] Prove that the line passing through $H$ and the circumcenter $O$ of triangle $ABC$ is a common tangent to $\omega_b$ and $\omega_c$. [/list] [i]Note:[/i] The orthocenter of a triangle is the intersection point of the three altitudes, whereas the circumcenter of a triangle is the center of the circumference passing through it's three vertices.

2020-21 IOQM India, 23

Tags: geometry , inradius , tangent circle

The incircle $\Gamma$ of a scalene triangle $ABC$ touches $BC$ at $D, CA$ at $E$ and $AB$ at $F$. Let $r_A$ be the radius of the circle inside $ABC$ which is tangent to $\Gamma$ and the sides $AB$ and $AC$. Define $r_B$ and $r_C$ similarly. If $r_A = 16, r_B = 25$ and $r_C = 36$, determine the radius of $\Gamma$.

2018 Greece Junior Math Olympiad, 4

Tags: geometry , tangent circle

Let $ABC$ with $AB<AC<BC$ be an acute angled triangle and $c$ its circumcircle. Let $D$ be the point diametrically opposite to $A$. Point $K$ is on $BD$ such that $KB=KC$. The circle $(K, KC)$ intersects $AC$ at point $E$. Prove that the circle $(BKE)$ is tangent to $c$.

1952 Kurschak Competition, 1

Tags: tangent circle , geometric inequalities , geometry

A circle $C$ touches three pairwise disjoint circles whose centers are collinear and none of which contains any of the others. Show that its radius must be larger than the radius of the middle of the three circles.

2003 Switzerland Team Selection Test, 8

Tags: circles , geometry , tangent circle

Let $A_1A_2A_3$ be a triangle and $\omega_1$ be a circle passing through $A_1$ and $A_2$. Suppose that there are circles $\omega_2,...,\omega_7$ such that: (a) $\omega_k$ passes through $A_k$ and $A_{k+1}$ for $k = 2,3,...,7$, where $A_i = A_{i+3}$, (b) $\omega_k$ and $\omega_{k+1}$ are externally tangent for $k = 1,2,...,6$. Prove that $\omega_1 = \omega_7$.

2023 JBMO Shortlist, G3

Tags: geometry , tangent circle , circumcircle

Let $A,B,C,D$ and $E$ be five points lying in this order on a circle, such that $AD=BC$. The lines $AD$ and $BC$ meet at a point $F$. The circumcircles of the triangles $CEF$ and $ABF$ meet again at the point $P$. Prove that the circumcircles of triangles $BDF$ and $BEP$ are tangent to each other.

2009 Ukraine Team Selection Test, 8

Tags: circles , fixed point , fixed , tangent circle , geometry

Two circles $\gamma_1, \gamma_2$ are given, with centers at points $O_1, O_2$ respectively. Select a point $K$ on circle $\gamma_2$ and construct two circles, one $\gamma_3$ that touches circle $\gamma_2$ at point $K$ and circle $\gamma_1$ at a point $A$, and the other $\gamma_4$ that touches circle $\gamma_2$ at point $K$ and circle $\gamma_1$ at a point $B$. Prove that, regardless of the choice of point K on circle $\gamma_2$, all lines $AB$ pass through a fixed point of the plane.

2020 Ukrainian Geometry Olympiad - April, 2

Tags: tangent circle , geometry , tangent , isosceles

Let $ABC$ be an isosceles triangle with $AB=AC$. Circle $\Gamma$ lies outside $ABC$ and touches line $AC$ at point $C$. The point $D$ is chosen on circle $\Gamma$ such that the circumscribed circle of the triangle $ABD$ touches externally circle $\Gamma$. The segment $AD$ intersects circle $\Gamma$ at a point $E$ other than $D$. Prove that $BE$ is tangent to circle $\Gamma$ .

Ukrainian TYM Qualifying - geometry, III.13

Tags: geometry , regular polygon , tangent circle

Inside the regular $n$ -gon $M$ with side $a$ there are $n$ equal circles so that each touches two adjacent sides of the polygon $M$ and two other circles. Inside the formed "star", which is bounded by arcs, these $n$ equal circles are reconstructed so that each touches the two adjacent circles built in the previous step, and two more newly built circles. This process will take $k$ steps. Find the area $S_n (k)$ of the "star", which is formed in the center of the polygon $M$. Consider the spatial analogue of this problem.

the 6th XMO, 5

Tags: geometry , tangent circle

As shown in the figure, $\odot O$ is the circumcircle of $\vartriangle ABC$, $\odot J$ is inscribed in $\odot O$ and is tangent to $AB$, $AC$ at points $D$ and E respectively, line segment $FG$ and $\odot O$ are tangent to point $A$, and $AF =AG=AD$, the circumscribed circle of $\vartriangle AFB$ intersects $\odot J$ at point $S$. Prove that the circumscribed circle of $\vartriangle ASG$ is tangent to $\odot J$. [img]https://cdn.artofproblemsolving.com/attachments/a/a/62d44e071ea9903ebdd68b43943ba1d93b4138.png[/img]

1984 Dutch Mathematical Olympiad, 1

Tags: geometry , tangent circle , angle bisector

The circles $C_1$ and $C_2$ with radii $r_1$ and $r_2$ touch the line $p$ at the point $P$. $C_1$ lies entirely within $C_2$. Line $q \perp p$ intersects $p$ at $S$ and touches $C_1$ at $R$. $q$ intersects $C_2$ at $M$ and $N$, where $N$ is between $R$ and $S$. a) Prove that line $PR$ bisects angle $\angle MPN$. b) Calculate the ratio $r_1 : r_2$ if line $PN$ bisects angle $\angle RPS$.

Denmark (Mohr) - geometry, 2007.4

Tags: circles , geometry , angle , tangent circle

The figure shows a $60^o$ angle in which are placed $2007$ numbered circles (only the first three are shown in the figure). The circles are numbered according to size. The circles are tangent to the sides of the angle and to each other as shown. Circle number one has radius $1$. Determine the radius of circle number $2007$. [img]https://1.bp.blogspot.com/-1bsLIXZpol4/Xzb-Nk6ospI/AAAAAAAAMWk/jrx1zVYKbNELTWlDQ3zL9qc_22b2IJF6QCLcBGAsYHQ/s0/2007%2BMohr%2Bp4.png[/img]

2001 239 Open Mathematical Olympiad, 3

Tags: circles , tangent circle , equal angles , geometry

The circles $ S_1 $ and $ S_2 $ intersect at points $ A $ and $ B $. Circle $ S_3 $ externally touches $ S_1 $ and $ S_2 $ at points $ C $ and $ D $ respectively. Let $ PQ $ be a chord cut by the line $ AB $ on circle $ S_3 $, and $ K $ be the midpoint of $ CD $. Prove that $ \angle PKC = \angle QKC $.