This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 280

2011 Israel National Olympiad, 4
Tags: geometry , equilateral triangle , tangent circle
Let $\alpha_1,\alpha_2,\alpha_3$ be three congruent circles that are tangent to each other. A third circle $\beta$ is tangent to them at points $A_1,A_2,A_3$ respectively. Let $P$ be a point on $\beta$ which is different from $A_1,A_2,A_3$. For $i=1,2,3$, let $B_i$ be the second intersection point of the line $PA_i$ with circle $\alpha_i$. Prove that $\Delta B_1B_2B_3$ is equilateral.

2016 Saudi Arabia BMO TST, 2
Tags: incenter , collinear , circumcircle , tangent circle , geometry
Let $ABC$ be a triangle and $I$ its incenter. The point $D$ is on segment $BC$ and the circle $\omega$ is tangent to the circumcirle of triangle $ABC$ but is also tangent to $DC, DA$ at $E, F$, respectively. Prove that $E, F$ and $I$ are collinear.

2020-IMOC, G1
Tags: circumcircle , tangent circle , geometry
Let $O$ be the circumcenter of triangle $ABC$. Choose a point $X$ on the circumcircle $\odot (ABC)$ such that $OX\parallel BC$. Assume that $\odot(AXO)$ intersects $AB, AC$ at $E, F$, respectively, and $OE, OF$ intersect $BC$ at $P, Q$, respectively. Furthermore, assume that $\odot(XP Q)$ and $\odot (ABC)$ intersect at $R$. Prove that $OR$ and $\odot (XP Q)$ are tangent to each other. (ltf0501)

2015 IMO Shortlist, G6
Tags: geometry , tangent circle
Let $ABC$ be an acute triangle with $AB > AC$. Let $\Gamma $ be its circumcircle, $H$ its orthocenter, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. Let $Q$ be the point on $\Gamma$ such that $\angle HQA = 90^{\circ}$ and let $K$ be the point on $\Gamma$ such that $\angle HKQ = 90^{\circ}$. Assume that the points $A$, $B$, $C$, $K$ and $Q$ are all different and lie on $\Gamma$ in this order. Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other. Proposed by Ukraine

2001 239 Open Mathematical Olympiad, 3
Tags: geometry , equal angles , circles , tangent circle
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 $.

1995 ITAMO, 5
Tags: tangent , 3d geometry , geometry , sphere , circles , tangent circle
Two non-coplanar circles in space are tangent at a point and have the same tangents at this point. Show that both circles lie on some sphere.

2006 Sharygin Geometry Olympiad, 10.3
Tags: tangent , geometry , locus , tangent circle
Given a circle and a point $P$ inside it, different from the center. We consider pairs of circles tangent to the given internally and to each other at point $P$. Find the locus of the points of intersection of the common external tangents to these circles.

2019 Abels Math Contest (Norwegian MO) Final, 3a
Tags: area , tangent circle , geometry
Three circles are pairwise tangent, with none of them lying inside another. The centres of the circles are the corners of a triangle with circumference $1$. What is the smallest possible value for the sum of the areas of the circles?

Estonia Open Senior - geometry, 1997.2.3
Tags: geometry , equal circles , circles , square , tangent circle
The figure shows a square and three circles of equal radius tangent to each other and square passes. Find the radius of the circles if the square length is $1$. [img]http://3.bp.blogspot.com/-iIjwupkz7DQ/XnrIRhKIJnI/AAAAAAAALhA/clERrIDqEtcujzvZk_qu975wsTjKaxCLQCK4BGAYYCw/s400/97%2Bestonia%2Bopen%2Bs2.3.png[/img]

1997 Slovenia Team Selection Test, 1
Tags: common tangent , tangent , geometry , tangent circle
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 $K_2$.

2016 Korea Winter Program Practice Test, 1
Tags: geometry , tangent circle
There is circle $\omega$ and $A, B$ on it. Circle $\gamma_1$ tangent to $\omega$ on $T$ and $AB$ on $D$. Circle $\gamma_2$ tangent to $\omega$ on $S$ and $AB$ on $E$. (like the figure below) Let $AB\cap TS=C$. Prove that $CA=CB$ iff $CD=CE$

2019 Ukraine Team Selection Test, 3
Tags: geometry , circumcircle , orthocenter , altitude , tangent circle
Given an acute triangle $ABC$ . It's altitudes $AA_1 , BB_1$ and $CC_1$ intersect at a point $H$ , the orthocenter of $\vartriangle ABC$. Let the lines $B_1C_1$ and $AA_1$ intersect at a point $K$, point $M$ be the midpoint of the segment $AH$. Prove that the circumscribed circle of $\vartriangle MKB_1$ touches the circumscribed circle of $\vartriangle ABC$ if and only if $BA1 = 3A1C$. (Bondarenko Mykhailo)

Novosibirsk Oral Geo Oly IX, 2021.4
Tags: geometry , square , tangent circle
A semicircle of radius $5$ and a quarter of a circle of radius $8$ touch each other and are located inside the square as shown in the figure. Find the length of the part of the common tangent, enclosed in the same square. [img]https://cdn.artofproblemsolving.com/attachments/f/2/010f501a7bc1d34561f2fe585773816f168e93.png[/img]

Denmark (Mohr) - geometry, 1992.2
Tags: geometry , right triangle , tangent circle
In a right-angled triangle, $a$ and $b$ denote the lengths of the two catheti. A circle with radius $r$ has the center on the hypotenuse and touches both catheti. Show that $\frac{1}{a}+\frac{1}{b}=\frac{1}{r}$.

2001 239 Open Mathematical Olympiad, 7
Tags: geometric inequality , circles , radii , tangent circle , geometry
The quadrangle $ ABCD $ contains two circles of radii $ R_1 $ and $ R_2 $ tangent externally. The first circle touches the sides of $ DA $,$ AB $ and $ BC $, moreover, the sides of $ AB $ at the point $ E $. The second circle touches sides $ BC $, $ CD $ and $ DA $, and sides $ CD $ at $ F $. Diagonals of the quadrangle intersect at $ O $. Prove that $ OE + OF \leq 2 (R_1 + R_2) $. (F. Bakharev, S. Berlov)

2021 Iran MO (3rd Round), 2
Tags: geometry , geometric transformation , reflection , circumcircle , tangent circle
Given an acute triangle $ABC$, let $AD$ be an altitude and $H$ the orthocenter. Let $E$ denote the reflection of $H$ with respect to $A$. Point $X$ is chosen on the circumcircle of triangle $BDE$ such that $AC\| DX$ and point $Y$ is chosen on the circumcircle of triangle $CDE$ such that $DY\| AB$. Prove that the circumcircle of triangle $AXY$ is tangent to that of $ABC$.

2023 Francophone Mathematical Olympiad, 3
Tags: geometry , circumcircle , tangent circle
Let $\Gamma$ and $\Gamma'$ be two circles with centres $O$ and $O'$, such that $O$ belongs to $\Gamma'$. Let $M$ be a point on $\Gamma'$, outside of $\Gamma$. The tangents to $\Gamma$ that go through $M$ touch $\Gamma$ in two points $A$ and $B$, and cross $\Gamma'$ again in two points $C$ and $D$. Finally, let $E$ be the crossing point of the lines $AB$ and $CD$. Prove that the circumcircles of the triangles $CEO'$ and $DEO'$ are tangent to $\Gamma'$.

1997 Estonia National Olympiad, 5
Tags: geometry , rectangle , circles , tangent circle
Six small circles of radius $1$ are drawn so that they are all tangent to a larger circle, and two of them are tangent to sides $BC$ and $AD$ of a rectangle $ABCD$ at their midpoints $K$ and $L$. Each of the remaining four small circles is tangent to two sides of the rectangle. The large circle is tangent to sides $AB$ and $CD$ of the rectangle and cuts the other two sides. Find the radius of the large circle. [img]https://cdn.artofproblemsolving.com/attachments/b/4/a134da78d709fe7162c48d6b5c40bd1016c355.png[/img]

2007 Silk Road, 2
Tags: geometry , incircle , excircle , tangent circle
Let $\omega$ be the incircle of triangle $ABC$ touches $BC$ at point $K$ . Draw a circle passing through points $B$ and $C$ , and touching $\omega$ at the point $S$ . Prove that $S K$ passes through the center of the exscribed circle of triangle $A B C$ , tangent to side $B C$ .

2018 Kyiv Mathematical Festival, 2
Tags: tangent circle , geometry
Let $M$ be the intersection point of the medians $AD$ and $BE$ of a right triangle $ABC$ ($\angle C=90^\circ$), $\omega_1$ and $\omega_2$ be the circumcircles of triangles $AEM$ and $CDM.$ It is known that the circles $\omega_1$ and $\omega_2$ are tangent. Find the ratio in which the circle $\omega_2$ divides $AC.$

Cono Sur Shortlist - geometry, 2005.G2
Tags: ratio , geometry , area , tangent circle
Find the ratio between the sum of the areas of the circles and the area of the fourth circle that are shown in the figure Each circle passes through the center of the previous one and they are internally tangent. [img]https://cdn.artofproblemsolving.com/attachments/d/2/29d2be270f7bcf9aee793b0b01c2ef10131e06.jpg[/img]

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 $c$.

2003 France Team Selection Test, 1
Tags: circumcircle , homothety , incenter , geometry , tangent circle
Let $B$ be a point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of $AC$ from $B$. Prove that the circumcentre of triangle $BCD$ lies on the circumcircle of triangle $ABC$.

1989 Greece Junior Math Olympiad, 3
Tags: tangent circle , geometry
Given a square $ABCD$ of side $a$, we consider the circle $\omega$, tangent to side $BC$ and to the two semicircles of diameters $AB$ and $CD$. Calculate the radius of circle $\omega$,

1937 Eotvos Mathematical Competition, 2
Tags: 3d geometry , tangent circle , geometry
Two circles in space are said to be tangent to each other if they have a corni-non tangent at the same point of tangency. Assume that there are three circles in space which are mutually tangent at three distinct points. Prove that they either alI lie in a plane or all lie on a sphere.