Found problems: 821
2013 IMAR Test, 3
The closure (interior and boundary) of a convex quadrangle is covered by four closed discs centered at each vertex of the quadrangle each. Show that three of these discs cover the closure of the triangle determined by their centers.
Estonia Open Senior - geometry, 2009.1.3
Three circles in a plane have the sides of a triangle as their diameters. Prove that there is a point that is in the interior of all three circles.
1974 Putnam, B1
Which configurations of five (not necessarily distinct) points $p_1 ,\ldots, p_5$ on the circle $x^2 +y^2 =1$ maximize the sum of the ten distances
$$\sum_{i<j} d(p_i, p_j)?$$
1997 Croatia National Olympiad, Problem 2
Consider a circle $k$ and a point $K$ in the plane. For any two distinct points $P$ and $Q$ on $k$, denote by $k'$ the circle through $P,Q$ and $K$. The tangent to $k'$ at $K$ meets the line $PQ$ at point $M$. Describe the locus of the points $M$ when $P$ and $Q$ assume all possible positions.
2013 Dutch IMO TST, 3
Fix a triangle $ABC$. Let $\Gamma_1$ the circle through $B$, tangent to edge in $A$. Let $\Gamma_2$ the circle through C tangent to edge $AB$ in $A$. The second intersection of $\Gamma_1$ and $\Gamma_2$ is denoted by $D$. The line $AD$ has second intersection $E$ with the circumcircle of $\vartriangle ABC$. Show that $D$ is the midpoint of the segment $AE$.
1986 All Soviet Union Mathematical Olympiad, 424
Two circumferences, with the distance $d$ between centres, intersect in points $P$ and $Q$ . Two lines are drawn through the point $A$ on the first circumference ($Q\ne A\ne P$) and points $P$ and $Q$ . They intersect the second circumference in the points $B$ and $C$ .
a) Prove that the radius of the circle, circumscribed around the triangle$ABC$ , equals $d$.
b) Describe the set of the new circle's centres, if thepoint $A$ moves along all the first circumference.
1950 Moscow Mathematical Olympiad, 173
On a chess board, the boundaries of the squares are assumed to be black. Draw a circle of the greatest possible radius lying entirely on the black squares.
2017 Korea Winter Program Practice Test, 1
Let $\gamma_1, \gamma_2, \gamma_3$ be mutually externally tangent circles and $\Gamma_1, \Gamma_2, \Gamma_3$ also be mutually externally tangent circles. For each $1 \le i \le 3$, $\gamma_i$ and $\Gamma_{i+1}$ are externally tangent at $A_i$, $\gamma_i$ and $\Gamma_{i+2}$ are externally tangent at $B_i$, and $\gamma_i$ and $\Gamma_i$ do not meet. Show that the six points $A_1, A_2, A_3, B_1, B_2, B_3$ lie on either a line or a circle.
2024 Mozambican National MO Selection Test, P2
On a sheet divided into squares, each square measuring $2cm$, two circles are drawn such that both circles are inscribed in a square as in the figure below. Determine the minimum distance between the two circles.
2015 Junior Balkan Team Selection Tests - Romania, 3
Let $ABC$ be an acute triangle , with $AB \neq AC$ and denote its orthocenter by $H$ . The point $D$ is located on the side $BC$ and the circumcircles of the triangles $ABD$ and $ACD$ intersects for the second time the lines $AC$ , respectively $AB$ in the points $E$ respectively $F$. If we denote by $P$ the intersection point of $BE$ and $CF$ then show that $HP \parallel BC$ if and only if $AD$ passes through the circumcenter of the triangle $ABC$.
2011 Sharygin Geometry Olympiad, 4
Given the circle of radius $1$ and several its chords with the sum of lengths $1$. Prove that one can be inscribe a regular hexagon into that circle so that its sides don’t intersect those chords.
2014 Rioplatense Mathematical Olympiad, Level 3, 5
In the segment $A C$ a point $B$ is taken. Construct circles $T_1, T_2$ and $T_3$ of diameters $A B, BC$ and $AC$ respectively. A line that passes through $B$ cuts $T_3$ in the points $P$ and $Q$, and the circles $T_1$ and $T_2$ respectively at points $R$ and $S$. Prove that $PR = Q S$.
2000 BAMO, 2
Let $ABC$ be a triangle with $D$ the midpoint of side $AB, E$ the midpoint of side $BC$, and $F$ the midpoint of side $AC$. Let $k_1$ be the circle passing through points $A, D$, and $F$, let $k_2$ be the circle passing through points $B, E$, and $D$, and let $k_3$ be the circle passing through $C, F$, and $E$. Prove that circles $k_1, k_2$, and $k_3$ intersect in a point.
2020 Ukrainian Geometry Olympiad - April, 3
The circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$, point $M$ is the midpoint of $AB$. On line $AB$ select points $S_1$ and $S_2$. Let $S_1X_1$ and $S_1Y_1$ be tangents drawn from $S_1$ to circle $\omega_1$, similarly $S_2X_2$ and $S_2Y_2$ are tangents drawn from $S_2$ to circles $\omega_2$. Prove that if the point $M$ lies on the line $X_1X_2$, then it also lies on the line $Y_1Y_2$.
2018 Azerbaijan Junior NMO, 4
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 $PA=\sqrt{PB\cdot QC}$
1997 Estonia National Olympiad, 4
Let be given $n\ge 3$ distinct points in the plane. Is it always possible to find a circle which passes through three of the points and contains none of the remaining points
(a) inside the circle.
(b) inside the circle or on its boundary?
1951 Poland - Second Round, 6
The given points are $ A $ and $ B $ and the circle $ k $. Draw a circle passing through the points $ A $ and $ B $ and defining, at the intersection with the circle $ k $, a common chord of a given length $ d $.
2019 Argentina National Olympiad Level 2, 3
Let $\Gamma$ be a circle of center $S$ and radius $r$ and let be $A$ a point outside the circle. Let $BC$ be a diameter of $\Gamma$ such that $B$ does not belong to the line $AS$ and consider the point $O$ where the perpendicular bisectors of triangle $ABC$ intersect, that is, the circumcenter of $ABC$. Determine all possible locations of point $O$ when $B$ varies in circle $\Gamma$.
2019 Israel National Olympiad, 7
In the plane points $A,B,C$ are marked in blue and points $P,Q$ are marked in red (no 3 marked points lie on a line, and no 4 marked points lie on a circle). A circle is called [b]separating[/b] if all points of one color are inside it, and all points of the other color are outside of it. Denote by $O$ the circumcenter of $ABC$ and by $R$ the circumradius of $ABC$.
Prove that [b]exactly one[/b] of the following holds:
[list]
[*] There exists a separating circle;
[*] There exists a point $X$ on the segment $PQ$ which also lies inside the triangle $ABC$, for which $PX\cdot XQ = R^2-OX^2$.
2019 BMT Spring, 13
Two circles $O_1$ and $O_2$ intersect at points $A$ and $B$. Lines $\overline{AC}$ and $\overline{BD}$ are drawn such that $C$ is on $O_1$ and $D$ is on $O_2$ and $\overline{AC} \perp \overline{AB}$ and $\overline{BD} \perp \overline{AB}$. If minor arc $AB= 45$ degrees relative to $O_1$ and minor arc $AB= 60$ degrees relative to $O_2$ and the radius of $O_2 = 10$, the area of quadrilateral $CADB$ can be expressed in simplest form as $a + b\sqrt{k} + c\sqrt{\ell}$. Compute $a + b + c + k +\ell$.
1994 Bulgaria National Olympiad, 1
Two circles $k_1(O_1,R)$ and $k_2(O_2,r)$ are given in the plane such that $R \ge \sqrt2 r$ and $$O_1O_2 =\sqrt{R^2 +r^2 - r\sqrt{4R^2 +r^2}}.$$ Let $A$ be an arbitrary point on $k_1$. The tangents from $A$ to $k_2$ touch $k_2$ at $B$ and $C$ and intersect $k_1$ again at $D$ and $E$, respectively. Prove that $BD \cdot CE = r^2$
2022 Turkey Team Selection Test, 4
We have three circles $w_1$, $w_2$ and $\Gamma$ at the same side of line $l$ such that $w_1$ and $w_2$ are tangent to $l$ at $K$ and $L$ and to $\Gamma$ at $M$ and $N$, respectively. We know that $w_1$ and $w_2$ do not intersect and they are not in the same size. A circle passing through $K$ and $L$ intersect $\Gamma$ at $A$ and $B$. Let $R$ and $S$ be the reflections of $M$ and $N$ with respect to $l$. Prove that $A, B, R, S$ are concyclic.
2015 India PRMO, 20
$20.$ The circle $\omega$ touches the circle $\Omega$ internally at point $P.$ The centre $O$ of $\Omega$ is outside $\omega.$ Let $XY$ be a diameter of $\Omega$ which is also tangent to $\omega.$ Assume $PY>PX.$ Let $PY$ intersect $\omega$ at $z.$ If $YZ=2PZ,$ what is the magnitude of $\angle{PYX}$ in degrees $?$
2016 Saint Petersburg Mathematical Olympiad, 4
$N> 4$ points move around the circle, each with a constant speed. For Any four of them have a moment in time when they all meet. Prove that is the moment when all the points meet.
1982 Bulgaria National Olympiad, Problem 2
Let $n$ unit circles be given on a plane. Prove that on one of the circles there is an arc of length at least $\frac{2\pi}n$ not intersecting any other circle.