Olympiad problems

2015 Romania Team Selection Tests, 2

Let $ABC$ be a triangle . Let $A'$ be the center of the circle through the midpoint of the side $BC$ and the orthogonal projections of $B$ and $C$ on the lines of support of the internal bisectrices of the angles $ACB$ and $ABC$ , respectively ; the points $B'$ and $C'$ are defined similarly . Prove that the nine-point circle of the triangle $ABC$ and the circumcircle of $A'B'C'$ are concentric.

2021 Indonesia TST, G

Tags: geometry , parallel , circles , kharkiv masters

The circles $k_1$ and $k_2$ intersect at points $A$ and $B$, and $k_1$ passes through the center $O$ of the circle $k_2$. The line $p$ intersects $k_1$ at the points $K ,O$ and $k_2$ at the points $L ,M$ so that $L$ lies between $K$ and $O$. The point $P$ is the projection of $L$ on the line $AB$. Prove that $KP$ is parallel to the median of triangle $ABM$ drawn from the vertex $M$.

2015 India Regional MathematicaI Olympiad, 4

Tags: combinatorics , combinatorial geometry , circles

Suppose $28$ objects are placed along a circle at equal distances. In how many ways can $3$ objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?

Mathley 2014-15, 5

Tags: geometry , cyclic , circles

A quadrilateral $ABCD$ is inscribed in a circle $(O)$. Another circle $(I)$ is tangent to the diagonals $AC, BD$ at $M, N$ respectively. Suppose that $MN$ meets $AB,CD$ at $P, Q$ respectively. The circumcircle of triangle $IMN$ meets the circumcircles of $IAB, ICD$ at $K, L$ respectively, which are distinct from $I$. Prove that the lines $PK, QL$, and $OI$ are concurrent. Tran Minh Ngoc, a student of Ho Chi Minh City College, Ho Chi Minh

2013 Dutch IMO TST, 3

Tags: tangent , geometry , circles , midpoint

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 $AE$.

Estonia Open Junior - geometry, 1999.2.3

Tags: circumcircle , geometry , circles , tangent circle

On the plane there are two non-intersecting circles with equal radii and with centres $O_1$ and $O_2$, line $s$ going through these centres, and their common tangent $t$. The third circle is tangent to these two circles in points $K$ and $L$ respectively, line $s$ in point $M$ and line $t$ in point $P$. The point of tangency of line $t$ and the first circle is $N$. a) Find the length of the segment $O_1O_2$. b) Prove that the points $M, K$ and $N$ lie on the same line

1995 Belarus Team Selection Test, 2

Tags: common tangent , geometry , circles , tangent circle

Circles $S,S_1,S_2$ are given in a plane. $S_1$ and $S_2$ touch each other externally, and both touch $S$ internally at $A_1$ and $A_2$ respectively. The common internal tangent to $S_1$ and $S_2$ meets $S$ at $P$ and $Q.$ Let $B_1$ and $B_2$ be the intersections of $PA_1$ and $PA_2$ with $S_1$ and $S_2$, respectively. Prove that $B_1B_2$ is a common tangent to $S_1,S_2$

1968 Spain Mathematical Olympiad, 3

Tags: geometry , square , circles

Given a square whose side measures $a$, consider the set of all points of its plane through which passes a circumference of radius whose circle contains to the quoted square. You are asked to prove that the contour of the figure formed by the points with this property is formed by arcs of circumference, and determine the positions, their centers, their radii and their lengths.

1962 Putnam, B4

Tags: coloring , circles

The euclidean plane is divided into regions by drawing a finite number of circles. Show that it is possible to color each of these regions either red or blue in such a way that no two adjacent regions have the same color.

2006 Hanoi Open Mathematics Competitions, 7

Tags: geometry , circles

On the circle $(O)$ of radius $15$ cm are given $2$ points $A, B$. The altitude $OH$ of the triangle $OAB$ intersect $(O)$ at $C$. What is $AC$ if $AB = 16$ cm?

2011 Greece JBMO TST, 4

Tags: geometry , circumcircle , collinear , circles

Let $ABC$ be an acute and scalene triangle with $AB<AC$, inscribed in a circle $c(O,R)$ (with center $O$ and radius $R$). Circle $c_1(A,AB)$ intersects side $BC$ at point $E$ and circle $c$ at point $F$. $EF$ intersects for the second time circle $c$ at point $D$ and side $AC$ at point $M$. $AD$ intersects $BC$ at point $K$. Circumcircle of triangle $BKD$ intersects $AB$ at point $L$ . Prove that points $K,L,M$ lie on a line parallel to $BF$.

1994 North Macedonia National Olympiad, 4

Tags: point , combinatorics , circles

$1994$ points from the plane are given so that any $100$ of them can be selected $98$ that can be rounded (some points may be at the border of the circle) with a diameter of $1$. Determine the smallest number of circles with radius $1$, sufficient to cover all $1994$

1971 IMO Shortlist, 4

Tags: length , geometry , circles , locus problems

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.

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 $\Gamma_3$.

2012 Bosnia and Herzegovina Junior BMO TST, 1

Tags: pentagon , geometry , circles

On circle $k$ there are clockwise points $A$, $B$, $C$, $D$ and $E$ such that $\angle ABE = \angle BEC = \angle ECD = 45^{\circ}$. Prove that $AB^2 + CE^2 = BE^2 + CD^2$

2019 Yasinsky Geometry Olympiad, p3

Tags: geometry , angle bisector , circles , tangent circle

Two circles $\omega_1$ and $\omega_2$ are tangent externally at the point $P$. Through the point $A$ of the circle $\omega_1$ is drawn a tangent to this circle, which intersects the circle $\omega_2$ at points $B$ and $C$ (see figure). Line $CP$ intersects again the circle $\omega_1$ to $D$. Prove that the $PA$ is a bisector of the angle $DPB$. [img]https://1.bp.blogspot.com/-nmKZGdBXfao/XOd51gRFuyI/AAAAAAAAKO0/EYo2SCW0eGcJsF64-Avo6w73ugkIIQ30ACK4BGAYYCw/s1600/Yasinsky%2B2019%2Bp2.png[/img]

2019 Tuymaada Olympiad, 2

Tags: circumcircle , circles , common tangent , geometry

A triangle $ABC$ with $AB < AC$ is inscribed in a circle $\omega$. Circles $\gamma_1$ and $\gamma_2$ touch the lines $AB$ and $AC$, and their centres lie on the circumference of $\omega$. Prove that $C$ lies on a common external tangent to $\gamma_1$ and $\gamma_2$.

Kyiv City MO Seniors 2003+ geometry, 2014.11.4.1.

Tags: geometry , circles , midpoint

Construct for the triangle $ABC$ a circle $S$ passing through the point $B$ and touching the line $CA$ at the point $A$, a circle $T$ passing through the point $C$ and touches the line $BA$ at the point $A$. The second intersection point of the circles $S$ and $T$ is denoted by $D$. The intersection point of the line $AD$ and the circumscribed circle $\Delta ABC$ is denoted by $E$. Prove that $D$ is the midpoint of the segment $AE$.

2016 Irish Math Olympiad, 4

Tags: geometry , equal angles , circles

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 $\angle AMP = \angle BMP$.

2015 Indonesia MO Shortlist, G2

Tags: geometry , circles , perpendicular , collinear , tangent circle

Two circles that are not equal are tangent externally at point $R$. Suppose point $P$ is the intersection of the external common tangents of the two circles. Let $A$ and $B$ are two points on different circles so that $RA$ is perpendicular to $RB$. Show that the line $AB$ passes through $P$.

2007 Sharygin Geometry Olympiad, 3

Tags: geometry , circumcenter , locus , locus problems , circumcircle , circles

Given two circles intersecting at points $P$ and $Q$. Let C be an arbitrary point distinct from $P$ and $Q$ on the former circle. Let lines $CP$ and $CQ$ intersect again the latter circle at points A and B, respectively. Determine the locus of the circumcenters of triangles $ABC$.

2003 Cuba MO, 2

Tags: geometry , circles , circumcircle , angle

Let $A$ be a point outside the circle $\omega$ . The tangents from $A$ touch the circle at $B$ and $C$. Let $P$ be an arbitrary point on extension of $AC$ towards $C$, $Q$ the projection of $C$ onto $PB$ and $E$ the second intersection point of the circumcircle of $ABP$ with the circle $\omega$ . Prove that $\angle PEQ = 2\angle APB$

Durer Math Competition CD Finals - geometry, 2011.C3

Tags: geometry , circles , cyclic quadrilateral

Given a circle with four circles that intersect in pairs as shown in the figure. The "internal" the points of intersection are $A, B, C$ and $D$, while the ‘outer’ points of intersection are $E, F, G$ and $H$. Prove that the quadrilateral $ABCD$ is cyclic if also the quadrilateral $EFGH$ is also cyclic. [img]https://cdn.artofproblemsolving.com/attachments/0/0/6a369c93e37eefd57775fd8586bdff393e1914.png[/img]

1997 Estonia National Olympiad, 5

Tags: geometry , circles , equilateral , tangent circle

There are six small circles in the figure with a radius of $1$ and tangent to a large circle and the sides of the $ABC$ of an equilateral triangle, where touch points are $K, L$ and $M$ respectively with the midpoints of sides $AB, BC$ and $AC$. Find the radius of the large circle and the side of the triangle $ABC$. [img]https://cdn.artofproblemsolving.com/attachments/3/0/f858dcc5840759993ea2722fd9b9b15c18f491.png[/img]

2019 Junior Balkan Team Selection Tests - Romania, 4

Tags: number , combinatorics , circles

The numbers from $1$ through $100$ are written in some order on a circle. We call a pair of numbers on the circle [i]good [/i] if the two numbers are not neighbors on the circle and if at least one of the two arcs they determine on the circle only contains numbers smaller then both of them. What may be the total number of good pairs on the circle.