Olympiad problems

I Soros Olympiad 1994-95 (Rus + Ukr), 9.6

Given a regular hexagon, whose sidelength is $ 1$ . What is the largest number of circles of radius $\frac{\sqrt3}{4}$ can be placed without overlapping inside such a hexagon? (Circles can touch each other and the sides of the hexagon.)

2014 Sharygin Geometry Olympiad, 2

Tags: circles , geometry

In a quadrilateral $ABCD$ angles $A$ and $C$ are right. Two circles with diameters $AB$ and $CD$ meet at points $X$ and $Y$ . Prove that line $XY$ passes through the midpoint of $AC$. (F. Nilov )

1974 Putnam, B1

Tags: circles , distance

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

2017 Balkan MO Shortlist, C4

Tags: combinatorics , minimum , radius , circles , combinatorial geometry

For any set of points $A_1, A_2,...,A_n$ on the plane, one defines $r( A_1, A_2,...,A_n)$ as the radius of the smallest circle that contains all of these points. Prove that if $n \ge 3$, there are indices $i,j,k$ such that $r( A_1, A_2,...,A_n)=r( A_i, A_j,A_k)$

1962 IMO Shortlist, 6

Tags: geometry , circles , trigonometry , isosceles , circumcircle

Consider an isosceles triangle. let $R$ be the radius of its circumscribed circle and $r$ be the radius of its inscribed circle. Prove that the distance $d$ between the centers of these two circle is \[ d=\sqrt{R(R-2r)} \]

1963 Bulgaria National Olympiad, Problem 3

Tags: circles , trapezoid , geometry

In the trapezium $ABCD$, a point $M$ is chosen on the non-base segment $AB$. Through the points $M,A,D$ and $M,B,C$ are drawn circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$. Prove that: (a) the second intersection point $N$ of $k_1$ and $k_2$ lies on the other non-base segment $CD$ or on its continuation; (b) the length of the line $O_1O_2$ doesn’t depend on the location of $M$ on $AB$; (c) the triangles $O_1MO_2$ and $DMC$ are similar. Find such a position of $M$ on $AB$ that makes $k_1$ and $k_2$ have the same radius.

1997 Estonia National Olympiad, 4

Tags: combinatorics , circles , combinatorial geometry

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?

2018 Tuymaada Olympiad, 2

Tags: circles , geometry , tangent

A circle touches the side $AB$ of the triangle $ABC$ at $A$, touches the side $BC$ at $P$ and intersects the side $AC$ at $Q$. The line symmetrical to $PQ$ with respect to $AC$ meets the line $AP$ at $X$. Prove that $PC=CX$. [i]Proposed by S. Berlov[/i]

1971 IMO Shortlist, 4

Tags: circles , geometry , length , 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.

2016 Poland - Second Round, 5

Tags: quadrilateral , cyclic quadrilateral , geometry , circles

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 $AC$.

2015 Romania Team Selection Tests, 2

Tags: circumcircle , geometry , euler circle , circles

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.

2012 IMAC Arhimede, 5

Tags: arc , combinatorics , circles , combinatorial geometry

On the circumference of a circle, there are $3n$ colored points that divide the circle on $3n$ arches, $n$ of which have lenght $1$, $n$ of which have length $2$ and the rest of them have length $3$ . Prove that there are two colored points on the same diameter of the circle.

2003 Switzerland Team Selection Test, 8

Tags: circles , geometry , tangent circle

Let $A_1A_2A_3$ be a triangle and $\omega_1$ be a circle passing through $A_1$ and $A_2$. Suppose that there are circles $\omega_2,...,\omega_7$ such that: (a) $\omega_k$ passes through $A_k$ and $A_{k+1}$ for $k = 2,3,...,7$, where $A_i = A_{i+3}$, (b) $\omega_k$ and $\omega_{k+1}$ are externally tangent for $k = 1,2,...,6$. Prove that $\omega_1 = \omega_7$.

2014 Czech-Polish-Slovak Junior Match, 4

Tags: geometry , perpendicular , equal area , area of a triangle , area , circles

Point $M$ is the midpoint of the side $AB$ of an acute triangle $ABC$. Circle with center $M$ passing through point $ C$, intersects lines $AC ,BC$ for the second time at points $P,Q$ respectively. Point $R$ lies on segment $AB$ such that the triangles $APR$ and $BQR$ have equal areas. Prove that lines $PQ$ and $CR$ are perpendicular.

2018 Azerbaijan Senior NMO, 3

Tags: geometry , tangent , angle bisector , circles

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 $PA=\sqrt{PB\cdot QC}$

2009 Ukraine Team Selection Test, 8

Tags: circles , fixed point , fixed , tangent circle , geometry

Two circles $\gamma_1, \gamma_2$ are given, with centers at points $O_1, O_2$ respectively. Select a point $K$ on circle $\gamma_2$ and construct two circles, one $\gamma_3$ that touches circle $\gamma_2$ at point $K$ and circle $\gamma_1$ at a point $A$, and the other $\gamma_4$ that touches circle $\gamma_2$ at point $K$ and circle $\gamma_1$ at a point $B$. Prove that, regardless of the choice of point K on circle $\gamma_2$, all lines $AB$ pass through a fixed point of the plane.

1998 Tournament Of Towns, 5

Tags: geometry , angle bisector , reflection , circles

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)

2012 Sharygin Geometry Olympiad, 7

Tags: geometry , circles , bijection

Consider a triangle $ABC$. The tangent line to its circumcircle at point $C$ meets line $AB$ at point $D$. The tangent lines to the circumcircle of triangle $ACD$ at points $A$ and $C$ meet at point $K$. Prove that line $DK$ bisects segment $BC$. (F.Ivlev)

2013 Balkan MO Shortlist, G4

Tags: circles , concurrency , geometry

Let $c(O, R)$ be a circle, $AB$ a diameter and $C$ an arbitrary point on the circle different than $A$ and $B$ such that $\angle AOC > 90^o$. On the radius $OC$ we consider point $K$ and the circle $c_1(K, KC)$. The extension of the segment $KB$ meets the circle $(c)$ at point $E$. From $E$ we consider the tangents $ES$ and $ET$ to the circle $(c_1)$. Prove that the lines $BE, ST$ and $AC$ are concurrent.

2001 Chile National Olympiad, 6

Tags: circles , geometry , tangential quadrilateral

Let $ C_1, C_2 $ be two circles of equal radius, disjoint, of centers $ O_1, O_2 $, such that $ C_1 $ is to the left of $ C_2 $. Let $ l $ be a line parallel to the line $ O_1O_2 $, secant to both circles. Let $ P_1 $ be a point of $ l $, to the left of $ C_1 $ and $ P_2 $ a point of $ l $, to the right of $ C_2 $ such that the tangents of $ P_1 $ to $ C_1 $ and of $ P_2 $ a $ C_2 $ form a quadrilateral. Show that there is a circle tangent to the four sides of said quadrilateral.

2019 Finnish National High School Mathematics Comp, 3

Tags: midpoint , geometry , cyclic quadrilateral , circles

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 $GH$.

Kyiv City MO Seniors 2003+ geometry, 2021.10.3

Tags: circles , equal segments , parallelogram , geometry

Circles $\omega_1$ and $\omega_2$ with centers at points $O_1$ and $O_2$ intersect at points $A$ and $B$. A point $C$ is constructed such that $AO_2CO_1$ is a parallelogram. An arbitrary line is drawn through point $A$, which intersects the circles $\omega_1$ and $\omega_2$ for the second time at points $X$ and $Y$, respectively. Prove that $CX = CY$. (Oleksii Masalitin)

2011 Sharygin Geometry Olympiad, 15

Tags: area , geometry , equal area , quadrilateral , circles , tangent

Given a circle with center $O$ and radius equal to $1$. $AB$ and $AC$ are the tangents to this circle from point $A$. Point $M$ on the circle is such that the areas of quadrilaterals $OBMC$ and $ABMC$ are equal. Find $MA$.

2015 FYROM JBMO Team Selection Test, 2

Tags: geometry , tangent , cyclic quadrilateral , circles

A circle $k$ with center $O$ and radius $r$ and a line $p$ which has no common points with $k$, are given. Let $E$ be the foot of the perpendicular from $O$ to $p$. Let $M$ be an arbitrary point on $p$, distinct from $E$. The tangents from the point $M$ to the circle $k$ are $MA$ and $MB$. If $H$ is the intersection of $AB$ and $OE$, then prove that $OH=\frac{r^2}{OE}$.

2019 Tuymaada Olympiad, 7

Tags: geometry , circles , tangent

A circle $\omega$ touches the sides $A$B and $BC$ of a triangle $ABC$ and intersects its side $AC$ at $K$. It is known that the tangent to $\omega$ at $K$ is symmetrical to the line $AC$ with respect to the line $BK$. What can be the difference $AK -CK$ if $AB = 9$ and $BC = 11$?