Olympiad problems

1998 Tournament Of Towns, 5

A circle with center $O$ is inscribed in an angle. Let $A$ be the reflection of $O$ across one side of the angle. Tangents to the circle from $A$ intersect the other side of the angle at points $B$ and $C$. Prove that the circumcenter of triangle $ABC$ lies on the bisector of the original angle. (I.Sharygin)

2007 Hanoi Open Mathematics Competitions, 5

Tags: geometry , circle

Suppose that $A,B,C,D$ are points on a circle, $AB$ is the diameter, $CD$ is perpendicular to $AB$ and meets $AB$ and meets $AB$ at $E , AB$ and $CD$ are integers and $AE - EB=\sqrt{3}$. Find $AE$?

2019 Argentina National Olympiad Level 2, 3

Tags: geometry , Circumcenter , Locus , circle

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 $\Gamma$.

2020 German National Olympiad, 1

Tags: geometry , geometry proposed , angles , circle , Maximal

Let $k$ be a circle with center $M$ and let $B$ be another point in the interior of $k$. Determine those points $V$ on $k$ for which $\measuredangle BVM$ becomes maximal.

1974 Putnam, B1

Tags: Putnam , distance , circle

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 $$\sum_{i<j} d(p_i, p_j)?$$

2006 Sharygin Geometry Olympiad, 24

Tags: geometry , Locus , 3D geometry , projections , Plane , circle , sphere

a) Two perpendicular rays are drawn through a fixed point $P$ inside a given circle, intersecting the circle at points $A$ and $B$. Find the geometric locus of the projections of $P$ on the lines $AB$. b) Three pairwise perpendicular rays passing through the fixed point $P$ inside a given sphere intersect the sphere at points $A, B, C$. Find the geometrical locus of the projections $P$ on the $ABC$ plane

1966 IMO Longlists, 28

Tags: geometry , circle , Locus , Locus problems , IMO Longlist , IMO Shortlist

In the plane, consider a circle with center $S$ and radius $1.$ Let $ABC$ be an arbitrary triangle having this circle as its incircle, and assume that $SA\leq SB\leq SC.$ Find the locus of [b]a.)[/b] all vertices $A$ of such triangles; [b]b.)[/b] all vertices $B$ of such triangles; [b]c.)[/b] all vertices $C$ of such triangles.

2019 Yasinsky Geometry Olympiad, p1

Tags: analytic geometry , circle

A circle with center at the origin and radius $5$ intersects the abscissa in points $A$ and $B$. Let $P$ a point lying on the line $x = 11$, and the point $Q$ is the intersection point of $AP$ with this circle. We know what is the $Q$ point is the midpoint of the $AP$. Find the coordinates of the point $P$.

2001 Bosnia and Herzegovina Team Selection Test, 1

Tags: ratio , geometry , circle , angles , arc

On circle there are points $A$, $B$ and $C$ such that they divide circle in ratio $3:5:7$. Find angles of triangle $ABC$

Kyiv City MO Seniors 2003+ geometry, 2005.11.2

Tags: geometry , isosceles , circle

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)

Denmark (Mohr) - geometry, 1991.5

Tags: geometry , combinatorial geometry , combinatorics , circle , points

Show that no matter how $15$ points are plotted within a circle of radius $2$ (circle border included), there will be a circle with radius $1$ (circle border including) which contains at least three of the $15$ points.

2013 Junior Balkan Team Selection Tests - Romania, 3

Tags: geometry , circle , Line

Consider a circle centered at $O$ with radius $r$ and a line $\ell$ not passing through $O$. A grasshopper is jumping to and fro between the points of the circle and the line, the length of each jump being $r$. Prove that there are at most $8$ points for the grasshopper to reach.

2004 Estonia National Olympiad, 1

Tags: geometry , circle , equal areas , chord

Inside a circle, point $K$ is taken such that the ray drawn from $K$ through the centre $O$ of the circle and the chord perpendicular to this ray passing through $K$ divide the circle into three pieces with equal area. Let $L$ be one of the endpoints of the chord mentioned. Does the inequality $\angle KOL < 75^o$ hold?

Geometry Mathley 2011-12, 4.2

Tags: perpendicular , circle , geometry

Let $ABC$ be a triangle. $(K)$ is an arbitrary circle tangent to the lines $AC,AB$ at $E, F$ respectively. $(K)$ cuts $BC$ at $M,N$ such that $N$ lies between $B$ and $M$. $FM$ intersects $EN$ at $I$. The circumcircles of triangles $IFN$ and $IEM$ meet each other at $J$ distinct from $I$. Prove that $IJ$ passes through $A$ and $KJ$ is perpendicular to $IJ$. Trần Quang Hùng

2004 Bosnia and Herzegovina Team Selection Test, 1

Tags: circle , collinear , touching , geometry

Circle $k$ with center $O$ is touched from inside by two circles in points $S$ and $T,$ respectively. Let those two circles intersect at points $M$ and $N$, such that $N$ is closer to line $ST$. Prove that $OM$ and $MN$ are perpendicular iff $S$, $N$ and $T$ are collinear

2011 Tournament of Towns, 7

Tags: points , combinatorial geometry , combinatorics , circle

$100$ red points divide a blue circle into $100$ arcs such that their lengths are all positive integers from $1$ to $100$ in an arbitrary order. Prove that there exist two perpendicular chords with red endpoints.

2004 Chile National Olympiad, 6

Tags: geometry , collinear , circle , convex quadrilateral

The $ AB, BC $ and $ CD $ segments of the polygon $ ABCD $ have the same length and are tangent to a circle $ S $, centered on the point $ O $. Let $ P $ be the point of tangency of $ BC $ with $ S $, and let $ Q $ be the intersection point of lines $ AC $ and $ BD $. Show that the point $ Q $ is collinear with the points $ P $ and $ O $.

1992 Austrian-Polish Competition, 5

Tags: geometry , Fixed point , fixed , circle , Product

Given a circle $k$ with center $M$ and radius $r$, let $AB$ be a fixed diameter of $k$ and let $K$ be a fixed point on the segment $AM$. Denote by $t$ the tangent of $k$ at $A$. For any chord $CD$ through $K$ other than $AB$, denote by $P$ and Q the intersection points of $BC$ and $BD$ with $t$, respectively. Prove that $AP\cdot AQ$ does not depend on $CD$.

1957 Putnam, A7

Tags: Putnam , circle , tangency

Each member of a set of circles in the $xy$-plane is tangent to the $x$-axis and no two of the circles intersect. Show that (a) the points of tangency can include all rational points on the axis. (b) the points of tangency cannot include all the irrational points.

1997 Abels Math Contest (Norwegian MO), 2b

Tags: isosceles , circle , Product , independent , geometry

Let $A,B,C$ be different points on a circle such that $AB = AC$. Point $E$ lies on the segment $BC$, and $D \ne A$ is the intersection point of the circle and line $AE$. Show that the product $AE \cdot AD$ is independent of the choice of $E$.

1978 Germany Team Selection Test, 6

Tags: combinatorics , lattice points , combinatorial geometry , circle , IMO Shortlist

A lattice point in the plane is a point both of whose coordinates are integers. Each lattice point has four neighboring points: upper, lower, left, and right. Let $k$ be a circle with radius $r \geq 2$, that does not pass through any lattice point. An interior boundary point is a lattice point lying inside the circle $k$ that has a neighboring point lying outside $k$. Similarly, an exterior boundary point is a lattice point lying outside the circle $k$ that has a neighboring point lying inside $k$. Prove that there are four more exterior boundary points than interior boundary points.

Swiss NMO - geometry, 2014.10

Tags: geometry , circle , bisects segment

Let $k$ be a circle with diameter $AB$. Let $C$ be a point on the straight line $AB$, so that $B$ between $A$ and $C$ lies. Let $T$ be a point on $k$ such that $CT$ is a tangent to $k$. Let $l$ be the parallel to $CT$ through $A$ and $D$ the intersection of $l$ and the perpendicular to $AB$ through $T$. Show that the line $DB$ bisects segment $CT$.

2007 All-Russian Olympiad, 4

Tags: algorithm , combinatorics , circle , Russia , game , trick

[i]A. Akopyan, A. Akopyan, A. Akopyan, I. Bogdanov[/i] A conjurer Arutyun and his assistant Amayak are going to show following super-trick. A circle is drawn on the board in the room. Spectators mark $2007$ points on this circle, after that Amayak removes one of them. Then Arutyun comes to the room and shows a semicircle, to which the removed point belonged. Explain, how Arutyun and Amayak may show this super-trick.

1966 IMO Shortlist, 16

Tags: geometry , Locus problems , Locus , circle , IMO Shortlist , IMO Longlist

We are given a circle $K$ with center $S$ and radius $1$ and a square $Q$ with center $M$ and side $2$. Let $XY$ be the hypotenuse of an isosceles right triangle $XY Z$. Describe the locus of points $Z$ as $X$ varies along $K$ and $Y$ varies along the boundary of $Q.$

1998 Bosnia and Herzegovina Team Selection Test, 4

Tags: circle , tangent , geometry , perpendicular

Circle $k$ with radius $r$ touches the line $p$ in point $A$. Let $AB$ be a dimeter of circle and $C$ an arbitrary point of circle distinct from points $A$ and $B$. Let $D$ be a foot of perpendicular from point $C$ to line $AB$. Let $E$ be a point on extension of line $CD$, over point $D$, such that $ED=BC$. Let tangents on circle from point $E$ intersect line $p$ in points $K$ and $N$. Prove that length of $KN$ does not depend from $C$