Found problems: 287
1987 Austrian-Polish Competition, 6
Let $C$ be a unit circle and $n \ge 1$ be a fixed integer. For any set $A$ of $n$ points $P_1,..., P_n$ on $C$ define $D(A) = \underset{d}{max}\, \underset{i}{min}\delta (P_i, d)$, where $d$ goes over all diameters of $C$ and $\delta (P, \ell)$ denotes the distance from point $P$ to line $\ell$. Let $F_n$ be the family of all such sets $A$. Determine $D_n = \underset{A\in F_n}{min} D(A)$ and describe all sets $A$ with $D(A) = D_n$.
2012 India Regional Mathematical Olympiad, 1
Let $ABCD$ be a unit square. Draw a quadrant of the a circle with $A$ as centre and $B,D$ as end points of the arc. Similarly, draw a quadrant of a circle with $B$ as centre and $A,C$ as end points of the arc. Inscribe a circle $\Gamma$ touching the arc $AC$ externally, the arc $BD$ externally and also touching the side $AD$. Find the radius of $\Gamma$.
2019 Switzerland - Final Round, 1
Let $A$ be a point and let k be a circle through $A$. Let $B$ and $C$ be two more points on $k$. Let $X$ be the intersection of the bisector of $\angle ABC$ with $k$. Let $Y$ be the reflection of $A$ wrt point $X$, and $D$ the intersection of the straight line $YC$ with $k$. Prove that point $D$ is independent of the choice of $B$ and $C$ on the circle $k$.
2016 Poland - Second Round, 5
Quadrilateral $ABCD$ is inscribed in circle. Points $P$ and $Q$ lie respectively on rays $AB^{\rightarrow}$ and $AD^{\rightarrow}$ such that $AP = CD$, $AQ = BC$. Show that middle point of line segment $PQ$ lies on the line $AC$.
2017 Yasinsky Geometry Olympiad, 3
Given circle $\omega$ and point $D$ outside this circle. Find the following points $A, B$ and $C$ on the circle $\omega$ so that the $ABCD$ quadrilateral is convex and has the maximum possible area. Justify your answer.
2014 AMC 8, 25
A straight one-mile stretch of highway, $40$ feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at $5$ miles per hour, how many hours will it take to cover the one-mile stretch?
Note: $1$ mile= $5280$ feet
[asy]size(10cm); pathpen=black; pointpen=black;
D(arc((-2,0),1,300,360));
D(arc((0,0),1,0,180));
D(arc((2,0),1,180,360));
D(arc((4,0),1,0,180));
D(arc((6,0),1,180,240));
D((-1.5,1)--(5.5,1));
D((-1.5,0)--(5.5,0),dashed);
D((-1.5,-1)--(5.5,-1));
[/asy]
$\textbf{(A) }\frac{\pi}{11}\qquad\textbf{(B) }\frac{\pi}{10}\qquad\textbf{(C) }\frac{\pi}{5}\qquad\textbf{(D) }\frac{2\pi}{5}\qquad \textbf{(E) }\frac{2\pi}{3}$
2012 German National Olympiad, 3
Let $ABC$ a triangle and $k$ a circle such that:
(1) The circle $k$ passes through $A$ and $B$ and touches the line $AC.$
(2) The tangent to $k$ at $B$ intersects the line $AC$ in a point $X\ne C.$
(3) The circumcircle $\omega$ of $BXC$ intersects $k$ in a point $Q\ne B.$
(4) The tangent to $\omega$ at $X$ intersects the line $AB$ in a point $Y.$
Prove that the line $XY$ is tangent to the circumcircle of $BQY.$
2007 Balkan MO Shortlist, G1
Let $\omega$ be a circle with center $O$ and let $A$ be a point outside $\omega$. The tangents from $A$ touch $\omega$ at points $B$, and $C$. Let $D$ be the point at which the line $AO$ intersects the circle such that $O$ is between $A$ and $D$. Denote by $X$ the orthogonal projection of $B$ onto $CD$, by $Y$ the midpoint of the segment $BX$ and by $Z$ the second point of intersection of the line $DY$ with $\omega$. Prove that $ZA$ and $ZC$ are perpendicular to each other.
1989 Austrian-Polish Competition, 4
Let $P$ be a convex polygon in the plane. Show that there exists a circle containing the entire polygon $P$ and having at least three adjacent vertices of $P$ on its boundary.
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?
1962 Czech and Slovak Olympiad III A, 4
Consider a circle $k$ with center $S$ and radius $r$. Let a point $A\neq S$ be given with $SA=d<r$. Consider a light ray emitted at point $A$, reflected at point $B\in k$, further reflected in point $C\in k$, which then passes through the original point $A$. Compute the sinus of convex angle $SAB$ in terms of $d,r$ and discuss conditions of solvability.
2019 Finnish National High School Mathematics Comp, 3
Let $ABCD$ be a cyclic quadrilateral whose side $AB$ is at the same time the diameter of the circle. The lines $AC$ and $BD$ intersect at point $E$ and the extensions of lines $AD$ and $BC$ intersect at point $F$. Segment $EF$ intersects the circle at $G$ and the extension of segment $EF$ intersects $AB$ at $H$. Show that if $G$ is the midpoint of $FH$, then $E$ is the midpoint of $GH$.
2012 Oral Moscow Geometry Olympiad, 5
Inside the circle with center $O$, points $A$ and $B$ are marked so that $OA = OB$. Draw a point $M$ on the circle from which the sum of the distances to points $A$ and $B$ is the smallest among all possible.
2016 Irish Math Olympiad, 4
Let $ABC$ be a triangle with $|AC| \ne |BC|$. Let $P$ and $Q$ be the intersection points of the line $AB$ with the internal and external angle bisectors at $C$, so that $P$ is between $A$ and $B$. Prove that if $M$ is any point on the circle with diameter $PQ$, then $\angle AMP = \angle BMP$.
2007 Chile National Olympiad, 2
Given a $\triangle ABC$, determine which is the circle with the smallest area that contains it.
1980 Austrian-Polish Competition, 9
Through the endpoints $A$ and $B$ of a diameter $AB$ of a given circle, the tangents $\ell$ and $m$ have been drawn. Let $C\ne A$ be a point on $\ell$ and let $q_1,q_2$ be two rays from $C$. Ray $q_i$ cuts the circle in $D_i$ and $E_i$ with $D_i$ between $C$ and $E_i, i = 1,2$. Rays $AD_1,AD_2,AE_1,AE_2$ meet $m$ in the respective points $M_1,M_2,N_1,N_2$. Prove that $M_1M_2 = N_1N_2$.
2012 Tournament of Towns, 2
One hundred points are marked inside a circle, with no three in a line. Prove that it is possible to connect the points in pairs such that all fifty lines intersect one another inside the circle.
2008 Greece JBMO TST, 1
Given a point $A$ that lies on circle $c(o,R)$ (with center $O$ and radius $R$). Let $(e)$ be the tangent of the circle $c$ at point $A$ and a line $(d)$ that passes through point $O$ and intersects $(e)$ at point $M$ and the circle at points $B,C$ (let $B$ lie between $O$ and $A$). If $AM = R\sqrt3$ , prove that
a) triangle $AMC$ is isosceles.
b) circumcenter of triangle $AMC$ lies on circle $c$ .
2010 Oral Moscow Geometry Olympiad, 6
Perpendicular bisectors of the sides $BC$ and $AC$ of an acute-angled triangle $ABC$ intersect lines $AC$ and $BC$ at points $M$ and $N$. Let point $C$ move along the circumscribed circle of triangle $ABC$, remaining in the same half-plane relative to $AB$ (while points $A$ and $B$ are fixed). Prove that line $MN$ touches a fixed circle.
1939 Moscow Mathematical Olympiad, 050
Given two points $A$ and $B$ and a circle, find a point $X$ on the circle so that points $C$ and $D$ at which lines $AX$ and $BX$ intersect the circle are the endpoints of the chord $CD$ parallel to a given line $MN$.
2012 India Regional Mathematical Olympiad, 7
On the extension of chord $AB$ of a circle centroid at $O$ a point $X$ is taken and tangents $XC$ and $XD$ to the circle are drawn from it with $C$ and $D$ lying on the circle, let $E$ be the midpoint of the line segment $CD$. If $\angle OEB = 140^o$ then determine with proof the magnitude of $\angle AOB$.
2009 Postal Coaching, 1
A circle $\Gamma$ and a line $\ell$ which does not intersect $\Gamma$ are given. Suppose $P, Q,R, S$ are variable points on circle $\Gamma$ such that the points $A = PQ\cap RS$ and $B = PS \cap QR$ lie on $\ell$. Prove that the circle on $AB$ as a diameter passes through two fixed points.
1973 IMO Shortlist, 2
Given a circle $K$, find the locus of vertices $A$ of parallelograms $ABCD$ with diagonals $AC \leq BD$, such that $BD$ is inside $K$.
2009 Oral Moscow Geometry Olympiad, 5
A treasure is buried at some point on a round island with a radius of $1$ km. On the coast of the island there is a mathematician with a device that indicates the direction to the treasure when the distance to the treasure does not exceed $500$ m. In addition, the mathematician has a map of the island, on which he can record all his movements, perform measurements and geometric constructions. The mathematician claims that he has an algorithm for how to get to the treasure after walking less than $4$ km. Could this be true?
(B. Frenkin)
2013 Saudi Arabia GMO TST, 4
In acute triangle $ABC$, points $D$ and $E$ are the feet of the perpendiculars from $A$ to $BC$ and $B$ to $CA$, respectively. Segment $AD$ is a diameter of circle $\omega$. Circle $\omega$ intersects sides $AC$ and $AB$ at $F$ and $G$ (other than $A$), respectively. Segment $BE$ intersects segments $GD$ and $GF$ at $X$ and $Y$ respectively. Ray $DY$ intersects side $AB$ at $Z$. Prove that lines $XZ$ and $BC$ are perpendicular