Olympiad problems

1971 IMO Longlists, 17

We are given two mutually tangent circles in the plane, with radii $r_1, r_2$. A line intersects these circles in four points, determining three segments of equal length. Find this length as a function of $r_1$ and $r_2$ and the condition for the solvability of the problem.

2010 Dutch IMO TST, 4

Tags: circles , equal segments , geometry , square

Let $ABCD$ be a square with circumcircle $\Gamma_1$. Let $P$ be a point on the arc $AC$ that also contains $B$. A circle $\Gamma_2$ touches $\Gamma_1$ in $P$ and also touches the diagonal $AC$ in $Q$. Let $R$ be a point on $\Gamma_2$ such that the line $DR$ touches $\Gamma_2$. Proof that $|DR| = |DA|$.

2019 Junior Balkan Team Selection Tests - Romania, 3

Tags: circles , geometry , perpendicular , collinear

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.

1962 Czech and Slovak Olympiad III A, 3

Tags: geometry , 3d geometry , line , locus , circles

Let skew lines $PM, QN$ be given such that $PM\perp PQ\perp QN$. Let a plane $\sigma\perp PQ$ containing the midpoint $O$ of segment $PQ$ be given and in it a circle $k$ with center $O$ and given radius $r$. Consider all segments $XY$ with endpoint $X, Y$ on lines $PM, QN$, respectively, which contain a point of $k$. Show that segments $XY$ have the same length. Find the locus of all such points $X$.

2009 Belarus Team Selection Test, 1

Tags: equal segments , circles , tangent , geometry , tangent circle

Two equal circles $S_1$ and $S_2$ meet at two different points. The line $\ell$ intersects $S_1$ at points $A,C$ and $S_2$ at points $B,D$ respectively (the order on $\ell$: $A,B,C,D$) . Define circles $\Gamma_1$ and $\Gamma_2$ as follows: both $\Gamma_1$ and $\Gamma_2$ touch $S_1$ internally and $S_2$ externally, both $\Gamma_1$ and $\Gamma_2$ line $\ell$, $\Gamma_1$ and $\Gamma_2$ lie in the different halfplanes relatively to line $\ell$. Suppose that $\Gamma_1$ and $\Gamma_2$ touch each other. Prove that $AB=CD$. I. Voronovich

1992 Poland - Second Round, 4

Tags: geometry , equal segments , circles

The circles $k_1$, $k_2$, $k_3$ are externally tangent: $k_1$ to $k_2$ at point $A$, $k_2$ to $k_3$ at point $B$, $k_3$ to $k_4$ at point $C$, $k_4$ to $k_1$ at point $D$. The lines $AB$ and $CD$ intersect at the point $S$. A line $ p $ is drawn through point $ S $, tangent to $ k_4 $ at point $ F $. Prove that $ |SE|=|SF| $.

III Soros Olympiad 1996 - 97 (Russia), 11.4

Tags: circles , geometry , tangent circle

There are four circles. The chord$ AB$ is drawn in the first one, and the distance from the midpoint of the smaller of the two formed arcs to $AB$ is equal to $1$. The second, third and fourth circles are located inside the larger segment and touch the chord $AB$. The second and fourth circles touch internally the first and externally the third. The sum of the radii of the last three circles is equal to the radius of the first circle. Find the radius of the third circle if it is known that the line passing through the centers of the first and third circles is not parallel to the line passing through the centers of the other two circles.

2019 Nigerian Senior MO Round 4, 2

Tags: midpoint , geometry , perpendicular , circles

Let $K,L, M$ be the midpoints of $BC,CA,AB$ repectively on a given triangle $ABC$. Let $\Gamma$ be a circle passing through $B$ and tangent to the circumcircle of $KLM$, say at $X$. Suppose that $LX$ and $BC$ meet at $\Gamma$ . Show that $CX$ is perpendicular to $AB$.

2017 Singapore MO Open, 1

Tags: midpoint , circles , incircle , circumcircle , geometry

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 $\vartriangle PQF$.

2011 Dutch IMO TST, 3

Tags: circles , tangent , geometry

Let $\Gamma_1$ and $\Gamma_2$ be two intersecting circles with midpoints respectively $O_1$ and $O_2$, such that $\Gamma_2$ intersects the line segment $O_1O_2$ in a point $A$. The intersection points of $\Gamma_1$ and $\Gamma_2$ are $C$ and $D$. The line $AD$ intersects $\Gamma_1$ a second time in $S$. The line $CS$ intersects $O_1O_2$ in $F$. Let $\Gamma_3$ be the circumcircle of triangle $AD$. Let $E$ be the second intersection point of $\Gamma_1$ and $\Gamma_3$. Prove that $O_1E$ is tangent to $\Gamma_3$.

2022 239 Open Mathematical Olympiad, 6

Tags: geometry , circles

On the side $BC$ of the rectangle $ABCD$, a point $P{}$ is marked so that $\angle APD = 90^\circ$. On the straight line $AD$, points $Q{}$ and $R{}$ are selected outside the segment $AD$ such that $AQ = BP$ and $CP = DR$. The circle $\omega$ passes through the points $Q, D$ and the circumcenter of the triangle $PDQ$. The circle $\gamma$ passes through the points $A, R$ and the circumcenter of the triangle $APR$. Prove that the radius of one of the circles touching the line $AD$ and the circles $\omega$ and $\gamma$ is $2AB$.

2007 Thailand Mathematical Olympiad, 3

Tags: circles , geometry , collinear

Two circles intersect at $X$ and $Y$ . The line through the centers of the circles intersect the first circle at $A$ and $C$, and intersect the second circle at $B$ and $D$ so that $A, B, C, D$ lie in this order. The common chord $XY$ cuts $BC$ at $P$, and a point $O$ is arbitrarily chosen on segment $XP$. Lines $CO$ and $BO$ are extended to intersect the first and second circles at $M$ and $N$, respectively. If lines $AM$ and $DN$ intersect at $Z$, prove that $X, Y$ and $Z$ lie on the same line.

2016 Hanoi Open Mathematics Competitions, 12

Tags: circles , geometry , fixed

Let $A$ be a point inside the acute angle $xOy$. An arbitrary circle $\omega$ passes through $O, A$, intersecting $Ox$ and $Oy$ at the second intersection $B$ and $C$, respectively. Let $M$ be the midpoint of $BC$. Prove that $M$ is always on a fixed line (when $\omega$ changes, but always goes through $O$ and $A$).

Estonia Open Junior - geometry, 1995.1.2

Tags: rectangle , geometry , circles

Two circles of equal radius intersect at two distinct points $A$ and $B$. Let their radii $r$ and their midpoints respectively be $O_1$ and $O_2$. Find the greatest possible value of the area of the rectangle $O_1AO_2B$.

2004 Tournament Of Towns, 4

Tags: circles , collinear , tangent , geometry

Two circles intersect in points $A$ and $B$. Their common tangent nearer $B$ touches the circles at points $E$ and $F$, and intersects the extension of $AB$ at the point $M$. The point $K$ is chosen on the extention of $AM$ so that $KM = MA$. The line $KE$ intersects the circle containing $E$ again at the point $C$. The line $KF$ intersects the circle containing $F$ again at the point $D$. Prove that the points $A, C$ and $D$ are collinear.

1982 All Soviet Union Mathematical Olympiad, 327

Tags: circles , area , geometry

Given two points $M$ and $K$ on the circumference with radius $r_1$ and centre $O_1$. The circumference with radius $r_2$ and centre $O_2$ is inscribed in $\angle MO_1K$ . Find the area of quadrangle $MO_1KO_2$ .

2017 Ukrainian Geometry Olympiad, 3

Tags: fixed , intersect , locus , locus problems , circles , geometry

Circles ${w}_{1},{w}_{2}$ intersect at points ${{A}_{1}} $ and ${{A}_{2}} $. Let $B$ be an arbitrary point on the circle ${{w}_{1}}$, and line $B{{A}_{2}}$ intersects circle ${{w}_{2}}$ at point $C$. Let $H$ be the orthocenter of $\Delta B{{A}_{1}}C$. Prove that for arbitrary choice of point $B$, the point $H$ lies on a certain fixed circle.

2019 Tournament Of Towns, 3

Tags: circles , geometry , angle bisector , disks , fixed , fixed point , coloring

Two equal non-intersecting wooden disks, one gray and one black, are glued to a plane. A triangle with one gray side and one black side can be moved along the plane so that the disks remain outside the triangle, while the colored sides of the triangle are tangent to the disks of the same color (the tangency points are not the vertices). Prove that the line that contains the bisector of the angle between the gray and black sides always passes through some fixed point of the plane. (Egor Bakaev, Pavel Kozhevnikov, Vladimir Rastorguev) (Senior version[url=https://artofproblemsolving.com/community/c6h2102856p15209040] here[/url])

1979 Bundeswettbewerb Mathematik, 2

Tags: geometry , locus , circles , incenter

A circle $k$ with center $M$ and radius $r$ is given. Find the locus of the incenters of all obtuse-angled triangles inscribed in $k$.

2018 Finnish National High School Mathematics Comp, 3

Tags: midpoint , geometry , chord , circles

The chords $AB$ and $CD$ of a circle intersect at $M$, which is the midpoint of the chord $PQ$. The points $X$ and $Y$ are the intersections of the segments $AD$ and $PQ$, respectively, and $BC$ and $PQ$, respectively. Show that $M$ is the midpoint of $XY$.

Geometry Mathley 2011-12, 13.1

Tags: fixed , circles , geometry , equal angles

Let $ABC$ be a triangle with no right angle, $E$ on the line $BC$ such that $\angle AEB = \angle BAC$ and $\Delta_A$ the perpendicular to $BC$ at $E$. Let the circle $\gamma$ with diameter $BC$ intersect $BA$ again at $D$. For each point $M$ on $\gamma$ ($M$ is distinct from $B$), the line $BM$ meets $\Delta_A$ at $M'$ and the line $AM$ meets $\gamma$ again at $M''$. (a) Show that $p(A) = AM' \times DM''$ is independent of the chosen $M$. (b) Keeping $B,C$ fixed, and let $A$ vary. Show that $\frac{p(A)}{d(A,\Delta_A)}$ is independent of $A$. Michel Bataille

1997 Romania National Olympiad, 1

Tags: geometry , circles , equal segments

Let $C_1,C_2,..., C_n$ , $(n\ge 3)$ be circles having a common point $M$. Three straight lines passing through $M$ intersect again the circles in $A_1, A_2,..., A_n$ ; $B_1,B_2,..., B_n$ and $X_1,X_2,..., X_n$ respectively. Prove that if $$A_1A_2 =A_2A_3 =...=A_{n-1}A_n$$ and $$B_1B_2 =B_2B_3 =...=B_{n-1}B_n$$ then $$X_1X_2 =X_2X_3 =...=X_{n-1}X_n.$$

2010 Sharygin Geometry Olympiad, 2

Tags: circles , geometry , trigonometry , trigonometric , circumcircle

Each of two equal circles $\omega_1$ and $\omega_2$ passes through the center of the other one. Triangle $ABC$ is inscribed into $\omega_1$, and lines $AC, BC$ touch $\omega_2$ . Prove that $cosA + cosB = 1$.

1977 IMO Shortlist, 2

Tags: combinatorics , combinatorial geometry , lattice points , circles

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.

2015 Saudi Arabia IMO TST, 2

Tags: orthocenter , circles , geometry , perpendicular

Let $ABC$ be a triangle with orthocenter $H$. Let $P$ be any point of the plane of the triangle. Let $\Omega$ be the circle with the diameter $AP$ . The circle $\Omega$ cuts $CA$ and $AB$ again at $E$ and $F$ , respectively. The line $PH$ cuts $\Omega$ again at $G$. The tangent lines to $\Omega$ at $E, F$ intersect at $T$. Let $M$ be the midpoint of $BC$ and $L$ be the point on $MG$ such that $AL$ and $MT$ are parallel. Prove that $LA$ and $LH$ are orthogonal. Lê Phúc Lữ