Found problems: 821
1945 Moscow Mathematical Olympiad, 105
A circle rolls along a side of an equilateral triangle. The radius of the circle is equal to the height of the triangle. Prove that the measure of the arc intercepted by the sides of the triangle on this circle is equal to $60^o$ at all times.
1966 IMO Shortlist, 14
What is the maximal number of regions a circle can be divided in by segments joining $n$ points on the boundary of the circle ?
[i]Posted already on the board I think...[/i]
1953 Moscow Mathematical Olympiad, 251
On a circle, distinct points $A_1, ... , A_{16}$ are chosen. Consider all possible convex polygons all of whose vertices are among $A_1, ... , A_{16}$ . These polygons are divided into $2$ groups, the first group comprising all polygons with $A_1$ as a vertex, the second group comprising the remaining polygons. Which group is more numerous?
2018 Israel National Olympiad, 6
In the corners of triangle $ABC$ there are three circles with the same radius. Each of them is tangent to two of the triangle's sides. The vertices of triangle $MNK$ lie on different sides of triangle $ABC$, and each edge of $MNK$ is also tangent to one of the three circles. Likewise, the vertices of triangle $PQR$ lie on different sides of triangle $ABC$, and each edge of $PQR$ is also tangent to one of the three circles (see picture below). Prove that triangles $MNK,PQR$ have the same inradius.
[img]https://i.imgur.com/bYuBabS.png[/img]
2011 Belarus Team Selection Test, 1
$AB$ and $CD$ are two parallel chords of a parabola. Circle $S_1$ passing through points $A,B$ intersects circle $S_2$ passing through $C,D$ at points $E,F$. Prove that if $E$ belongs to the parabola, then $F$ also belongs to the parabola.
I.Voronovich
1995 Denmark MO - Mohr Contest, 5
In the plane, six circles are given so that none of the circles contain one the center of the other. Show that there is no point that lies in all the circles.
1968 IMO Shortlist, 19
We are given a fixed point on the circle of radius $1$, and going from this point along the circumference in the positive direction on curved distances $0, 1, 2, \ldots $ from it we obtain points with abscisas $n = 0, 1, 2, .\ldots$ respectively. How many points among them should we take to ensure that some two of them are less than the distance $\frac 15$ apart ?
2017 Thailand Mathematical Olympiad, 10
A lattice point is defined as a point on the plane with integer coordinates. Show that for all positive integers $n$, there is a circle on the plane with exactly n lattice points in its interior (not including its boundary).
Geometry Mathley 2011-12, 8.4
Let $ABC$ a triangle inscribed in a circle $(O)$ with orthocenter $H$. Two lines $d_1$ and $d_2$ are mutually perpendicular at $H$. Let $d_1$ meet $BC,CA,AB$ at $X_1, Y_1,Z_1$ respectively. Let $A_1B_1C_1$ be a triangle formed by the line through $X_1$ perpendicular to $BC$, the line through $Y_1$ perpendicular to CA, the line through $Z_1$ perpendicular perpendicular to $AB$. Triangle $A_2B_2C_2$ is defined in the same manner. Prove that the circumcircles of triangles $A_1B_1C_1$ and $A_2B_2C_2$ touch each other at a point on $(O)$.
Nguyễn Văn Linh
1924 Eotvos Mathematical Competition, 3
Let $A$, $B$, and $C$ be three given points in the plane; construct three cirdes, $k_1$, $k_2$, and $k_3$, such that $k_2$ and $k_3$ have a common tangent at $A$, $k_3$ and $k_1$ at $B$, and $k_1$ and $k_2$ at $C$.
2010 Sharygin Geometry Olympiad, 4
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.
2005 Federal Math Competition of S&M, Problem 4
Inside a circle $k$ of radius $R$ some round spots are made. The area of each spot is $1$. Every radius of circle $k$, as well as every circle concentric with $k$, meets in no more than one spot. Prove that the total area of all the spots is less than
$$\pi\sqrt R+\frac12R\sqrt R.$$
2017 Singapore MO Open, 1
The incircle of $\vartriangle ABC$ touches the sides $BC,CA,AB$ at $D,E,F$ respectively. A circle through $A$ and $B$ encloses $\vartriangle ABC$ and intersects the line $DE$ at points $P$ and $Q$. Prove that the midpoint of $AB$ lies on the circumircle of $\vartriangle PQF$.
1954 Poland - Second Round, 1
The cross-section of a ball bearing consists of two concentric circles $ C $ and $ C_1 $, between which there are $ n $ small circles $ k_1, k_2, \ldots, k_n $, each of which is tangent to the two adjacent circles and to both circles $ C $ and $ C_1 $. Given the radius $ r $ of the inner circle $ C $ and a natural number $ n $, calculate the radius $ x $ of circle $ C_2 $ passing through the points of tangency of circles $ k_1, k_2, \ldots, k_n $ and the sum $ s $ of the lengths of the arcs of circles $ k_1, k_2, \ldots, k_n $ that lie outside circle $ C_2 $.
1980 IMO Longlists, 10
Two circles $C_{1}$ and $C_{2}$ are (externally or internally) tangent at a point $P$. The straight line $D$ is tangent at $A$ to one of the circles and cuts the other circle at the points $B$ and $C$. Prove that the straight line $PA$ is an interior or exterior bisector of the angle $\angle BPC$.
2020 Centroamerican and Caribbean Math Olympiad, 4
Consider a triangle $ABC$ with $BC>AC$. The circle with center $C$ and radius $AC$ intersects the segment $BC$ in $D$. Let $I$ be the incenter of triangle $ABC$ and $\Gamma$ be the circle that passes through $I$ and is tangent to the line $CA$ at $A$. The line $AB$ and $\Gamma$ intersect at a point $F$ with $F \neq A$. Prove that $BF=BD$.
2014 Contests, 2
Let $ABCD$ be an inscribed quadrilateral in a circle $c(O,R)$ (of circle $O$ and radius $R$). With centers the vertices $A,B,C,D$, we consider the circles $C_{A},C_{B},C_{C},C_{D}$ respectively, that do not intersect to each other . Circle $C_{A}$ intersects the sides of the quadrilateral at points $A_{1} , A_{2}$ , circle $C_{B}$ intersects the sides of the quadrilateral at points $B_{1} , B_{2}$ , circle $C_{C}$ at points $C_{1} , C_{2}$ and circle $C_{D}$ at points $C_{1} , C_{2}$ . Prove that the quadrilateral defined by lines $A_{1}A_{2} , B_{1}B_{2} , C_{1}C_{2} , D_{1}D_{2}$ is cyclic.
1975 Bulgaria National Olympiad, Problem 4
In the plane are given a circle $k$ with radii $R$ and the points $A_1,A_2,\ldots,A_n$, lying on $k$ or outside $k$. Prove that there exist infinitely many points $X$ from the given circumference for which
$$\sum_{i=1}^n A_iX^2\ge2nR^2.$$
Does there exist a pair of points on different sides of some diameter, $X$ and $Y$ from $k$, such that
$$\sum_{i=1}^n A_iX^2\ge2nR^2\text{ and }\sum_{i=1}^n A_iY^2\ge2nR^2?$$
[i]H. Lesov[/i]
Kyiv City MO Seniors 2003+ geometry, 2005.11.2
A circle touches the sides $AC$ and $AB$ of the triangle $ABC $ at the points ${{B}_ {1}} $ and ${{C}_ {1}}$ respectively. The segments $B {{B} _ {1}} $ and $C {{C} _ {1}}$ are equal. Prove that the triangle $ABC $ is isosceles.
(Timoshkevich Taras)
2012 Polish MO Finals, 3
Triangle $ABC$ with $AB = AC$ is inscribed in circle $o$. Circles $o_1$ and $o_2$ are internally tangent to circle $o$ in points $P$ and $Q$, respectively, and they are tangent to segments $AB$ and $AC$, respectively, and they are disjoint with the interior of triangle $ABC$. Let $m$ be a line tangent to circles $o_1$ and $o_2$, such that points $P$ and $Q$ lie on the opposite side than point $A$. Line $m$ cuts segments $AB$ and $AC$ in points $K$ and $L$, respectively. Prove, that intersection point of lines $PK$ and $QL$ lies on bisector of angle $BAC$.
2006 Junior Balkan Team Selection Tests - Romania, 2
Let $C (O)$ be a circle (with center $O$ ) and $A, B$ points on the circle with $\angle AOB = 90^o$. Circles $C_1 (O_1)$ and $C_2 (O_2)$ are tangent internally with circle $C$ at $A$ and $B$, respectively, and, also, are tangent to each other. Consider another circle $C_3 (O_3)$ tangent externally to the circles $C_1, C_2$ and tangent internally to circle $C$, located inside angle $\angle AOB$. Show that the points $O, O_1, O_2, O_3$ are the vertices of a rectangle.
2006 Estonia Team Selection Test, 4
The side $AC$ of an acute triangle $ABC$ is the diameter of the circle $c_1$ and side $BC$ is the diameter of the circle $c_2$. Let $E$ be the foot of the altitude drawn from the vertex $B$ of the triangle and $F$ the foot of the altitude drawn from the vertex $A$. In addition, let $L$ and $N$ be the points of intersection of the line $BE$ with the circle $c_1$ (the point $L$ lies on the segment $BE$) and the points of intersection of $K$ and $M$ of line $AF$ with circle $c_2$ (point $K$ is in section $AF$). Prove that $K LM N$ is a cyclic quadrilateral.
2019 Junior Balkan Team Selection Tests - Romania, 3
A circle with center $O$ is internally tangent to two circles inside it at points $S$ and $T$. Suppose the two circles inside intersect at $M$ and $N$ with $N$ closer to $ST$. Show that $OM$ and $MN$ are perpendicular if and only if $S,N, T$ are collinear.
2021 Brazil Team Selection Test, 3
Let $ABC$ be an acute triangle with $AC>CB$ and let $M$ be the midpoint of side $AB$. Denote by $Q$ the midpoint of the big arc $AB$ which cointais $C$ and by $B_1$ the point inside $AC$ such that $BC=CB_1$. $B_1Q$ touches $BC$ in $E$ and $K$ is the intersection of $(BB_1M)$ and $(ABC)$. Prove that $KC$ bissects $B_1E$.
2017 India PRMO, 27
Let $\Omega_1$ be a circle with centre $O$ and let $AB$ be diameter of $\Omega_1$. Let $P$ be a point on the segment $OB$ different from $O$. Suppose another circle $\Omega_2$ with centre $P$ lies in the interior of $\Omega_1$. Tangents are drawn from $A$ and $B$ to the circle $\Omega_2$ intersecting $\Omega_1$ again at $A_1$ and B1 respectively such that $A_1$ and $B_1$ are on the opposite sides of $AB$. Given that $A_1 B = 5, AB_1 = 15$ and $OP = 10$, find the radius of $\Omega_1$.