Olympiad problems

2019 Nigerian Senior MO Round 4, 2

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 $AB$.

1976 All Soviet Union Mathematical Olympiad, 232

Tags: circle , combinatorics

$n$ numbers are written down along the circumference. Their sum equals to zero, and one of them equals $1$. a) Prove that there are two neighbours with their difference not less than $n/4$. b) Prove that there is a number that differs from the arithmetic mean of its two neighbours not less than on $8/(n^2)$. c) Try to improve the previous estimation, i.e what number can be used instead of $8$? d) Prove that for $n=30$ there is a number that differs from the arithmetic mean of its two neighbours not less than on $2/113$, give an example of such $30$ numbers along the circumference, that not a single number differs from the arithmetic mean of its two neighbours more than on $2/113$.

1999 Denmark MO - Mohr Contest, 1

Tags: conics , parabola , circle , tangent

In a coordinate system, a circle with radius $7$ and center is on the y-axis placed inside the parabola with equation $y = x^2$ , so that it just touches the parabola in two points. Determine the coordinate set for the center of the circle.

1975 Chisinau City MO, 89

Tags: Cyclic , circle , geometry

A closed line on a plane is such that any quadrangle inscribed in it has the sum of opposite angles equal to $180^o$. Prove that this line is a circle.

2017 Balkan MO Shortlist, C4

Tags: combinatorics , minimum , radius , circle , 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 $r( A_1, A_2,...,A_n)=r( A_i, A_j,A_k)$

1998 Austrian-Polish Competition, 6

Tags: geometry , parallel , concurrency , concurrent , circle

Different points $A,B,C,D,E,F$ lie on circle $k$ in this order. The tangents to $k$ in the points $A$ and $D$ and the lines $BF$ and $CE$ have a common point $P$. Prove that the lines $AD,BC$ and $EF$ also have a common point or are parallel.

2008 Postal Coaching, 2

Tags: geometry , circle , ratio

Let $ABC$ be an equilateral triangle, and let $K, L,M$ be points respectively on $BC, CA, AB$ such that $BK/KC = CL/LA = AM/MB =\lambda $. Find all values of $\lambda$ such that the circle with $BC$ as a diameter completely covers the triangle bounded by the lines $AK,BL,CM$.

2008 Princeton University Math Competition, B5

Tags: circle

Two externally tangent circles have radius $2$ and radius $3$. Two lines are drawn, each tangent to both circles, but not at the point where the circles are tangent to each other. What is the area of the quadrilateral whose vertices are the four points of tangency between the circles and the lines?

Ukrainian TYM Qualifying - geometry, 2015.18

Tags: areas , circle , geometry , equal areas , Ukrainian TYM

Is it possible to divide a circle by three chords, different from diameters, into several equal parts?

Estonia Open Senior - geometry, 2010.1.4

Tags: geometry , circle , orthocenter , incenter , Circumcenter , isosceles

Circle $c$ passes through vertices $A$ and $B$ of an isosceles triangle $ABC$, whereby line $AC$ is tangent to it. Prove that circle $c$ passes through the circumcenter or the incenter or the orthocenter of triangle $ABC$.

2016 Germany Team Selection Test, 1

Tags: Concyclic , geometry unsolved , geometry , circle , circles

The two circles $\Gamma_1$ and $\Gamma_2$ with the midpoints $O_1$ resp. $O_2$ intersect in the two distinct points $A$ and $B$. A line through $A$ meets $\Gamma_1$ in $C \neq A$ and $\Gamma_2$ in $D \neq A$. The lines $CO_1$ and $DO_2$ intersect in $X$. Prove that the four points $O_1,O_2,B$ and $X$ are concyclic.

2011 Oral Moscow Geometry Olympiad, 3

Tags: geometry , trapezoid , concurrency , concurrent , circle

A non-isosceles trapezoid $ABCD$ ($AB // CD$) is given. An arbitrary circle passing through points $A$ and $B$ intersects the sides of the trapezoid at points $P$ and $Q$, and the intersect the diagonals at points $M$ and $N$. Prove that the lines $PQ, MN$ and $CD$ are concurrent.

Geometry Mathley 2011-12, 4.1

Tags: distance , circle , geometry

Five points $K_i, i = 1, 2, 3, 4$ and $P$ are chosen arbitrarily on the same circle. Denote by $P(i, j)$ the distance from $P$ to the line passing through $K_i$ and $K_j$ . Prove that $$P(1, 2)P(3, 4) = P(1, 4)P(2, 3) = P(1, 3)P(2, 4)$$ Bùi Quang Tuấn

1991 Denmark MO - Mohr Contest, 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.

1979 IMO Shortlist, 24

Tags: geometry , circle , tangency , Isosceles Triangle , IMO Shortlist , IMO Longlist

A circle $C$ with center $O$ on base $BC$ of an isosceles triangle $ABC$ is tangent to the equal sides $AB,AC$. If point $P$ on $AB$ and point $Q$ on $AC$ are selected such that $PB \times CQ = (\frac{BC}{2})^2$, prove that line segment $PQ$ is tangent to circle $C$, and prove the converse.

2014 Sharygin Geometry Olympiad, 7

Tags: geometry , circle

Two points on a circle are joined by a broken line shorter than the diameter of the circle. Prove that there exists a diameter which does not intersect this broken line. (Folklor )

2015 India Regional MathematicaI Olympiad, 4

Tags: combinatorics , combinatorial geometry , circle

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?

2011 Junior Balkan Team Selection Tests - Moldova, 7

Tags: geometry , rectangle , circle , collinear

In the rectangle $ABCD$ with $AB> BC$, the perpendicular bisecotr of $AC$ intersects the side $CD$ at point $E$. The circle with the center at point $E$ and the radius $AE$ intersects again the side $AB$ at point $F$. If point $O$ is the orthogonal projection of point $C$ on the line $EF$, prove that points $B, O$ and $D$ are collinear.

2012 IMAC Arhimede, 5

Tags: combinatorics , combinatorial geometry , arc , circle

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.

2012 Bosnia and Herzegovina Junior BMO TST, 1

Tags: circle , pentagon , geometry

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 $AB^2 + CE^2 = BE^2 + CD^2$

2025 Kosovo National Mathematical Olympiad`, P4

Tags: circle , concurrency , Lemma , points in sides , Cevians , Pop , geometry

Let $ABC$ be a given triangle. Let $A_1$ and $A_2$ be points on the side $BC$. Let $B_1$ and $B_2$ be points on the side $CA$. Let $C_1$ and $C_2$ be points on the side $AB$. Suppose that the points $A_1,A_2,B_1,B_2,C_1$ and $C_2$ lie on a circle. Prove that the lines $AA_1, BB_1$ and $CC_1$ are concurrent if and only if $AA_2, BB_2$ and $CC_2$ are concurrent.

2004 Tournament Of Towns, 5

Tags: parabola , common tangent , circle , geometry , tangent , Tangents

The parabola $y = x^2$ intersects a circle at exactly two points $A$ and $B$. If their tangents at $A$ coincide, must their tangents at $B$ also coincide?

Swiss NMO - geometry, 2015.4

Tags: geometry , circle , construction

Given a circle $k$ and two points $A$ and $B$ outside the circle. Specify how to can construct a circle with a compass and ruler, so that $A$ and $B$ lie on that circle and that circle is tangent to $k$.

1992 All Soviet Union Mathematical Olympiad, 563

Tags: reflection , symmetry , equidistant , geometry , circle

$A$ and $B$ lie on a circle. $P$ lies on the minor arc $AB$. $Q$ and $R$ (distinct from $P$) also lie on the circle, so that $P$ and $Q$ are equidistant from $A$, and $P$ and $R$ are equidistant from $B$. Show that the intersection of $AR$ and $BQ$ is the reflection of $P$ in $AB$.

Swiss NMO - geometry, 2006.2

Tags: geometry , Equilateral , circle

Let $ABC$ be an equilateral triangle and let $D$ be an inner point of the side $BC$. A circle is tangent to $BC$ at $D$ and intersects the sides $AB$ and $AC$ in the inner points $M, N$ and $P, Q$ respectively. Prove that $|BD| + |AM| + |AN| = |CD| + |AP| + |AQ|$.