Olympiad problems

2014 Czech-Polish-Slovak Junior Match, 4

Tags: area of a triangle , geometry , perpendicular , circle , areas , equal areas

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.

Kyiv City MO Seniors 2003+ geometry, 2016.10.4

Tags: geometry , Locus , circle , collinear

On the circle with diameter $AB$, the point $M$ was selected and fixed. Then the point ${{Q} _ {i}}$ is selected, for which the chord $M {{Q} _ {i}}$ intersects $AB$ at the point ${{K} _ {i}}$ and thus $ \angle M {{K} _ {i}} B <90 {} ^ \circ$. A chord that is perpendicular to $AB$ and passes through the point ${{K} _ {i}}$ intersects the line $B {{Q} _ {i}}$ at the point ${{P } _ {i}}$. Prove that the points ${{P} _ {i}}$ in all possible choices of the point ${{Q} _ {i}}$ lie on the same line. (Igor Nagel)

2008 Postal Coaching, 5

Tags: tangent , geometry , semicircle , parallel , circle

Let $\omega$ be the semicircle on diameter $AB$. A line parallel to $AB$ intersects $\omega$ at $C$ and $D$ so that $B$ and $C$ lie on opposite sides of $AD$. The line through $C$ parallel to $AD$ meets $\omega$ again in $E$. Lines $BE$ and $CD$ meet in $F$ and the line through $F$ parallel to $AD$ meets $AB$ in $P$. Prove that $PC$ is tangent to $\omega$.

2020 HK IMO Preliminary Selection Contest, 6

Tags: geometry , circle , circumcircle

In $\Delta ABC$, $AB=6$, $BC=7$ and $CA=8$. Let $D$ be the mid-point of minor arc $AB$ on the circumcircle of $\Delta ABC$. Find $AD^2$

1996 Tournament Of Towns, (507) 4

Tags: geometry , circle , rhombus

A circle cuts each side of a rhombus twice thus dividing each side into three segments. Let us go around the perimeter of the rhombus clockwise beginning at a vertex and paint these segments successively in red, white and blue. Prove that the sum of lengths of the blue segments equals that of the red ones. (V Proizvolov)

2019 Yasinsky Geometry Olympiad, p3

Tags: geometry , tangent circles , angle bisector , 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]

2022 Turkey EGMO TST, 5

Tags: geometry , circle , Tangents , lines

We are given three points $A,B,C$ on a semicircle. The tangent lines at $A$ and $B$ to the semicircle meet the extension of the diameter at points $M,N$ respectively. The line passing through $A$ that is perpendicular to the diameter meets $NC$ at $R$, and the line passing through $B$ that is perpendicular to the diameter meets $MC$ at $S$. If the line $RS$ meets the extension of the diameter at $Z$, prove that $ZC$ is tangent to the semicircle.

1984 IMO, 1

Tags: geometry , convex quadrilateral , circle , parallel , diameter , IMO , IMO 1984

Let $ABCD$ be a convex quadrilateral with the line $CD$ being tangent to the circle on diameter $AB$. Prove that the line $AB$ is tangent to the circle on diameter $CD$ if and only if the lines $BC$ and $AD$ are parallel.

1926 Eotvos Mathematical Competition, 3

Tags: geometry , circles , circle

The circle $k'$ rolls along the inside of circle $k$, the radius of $k$ is twice the radius of $k'$. Describe the path of a point on $k$..

2003 Korea Junior Math Olympiad, 3

Tags: geometry , ratio , circle

Consider a triangle $ABC$, inscribed in $O$ and $\angle A < \angle B$. Some point $P$ outside the circle satisfies $$\angle A=\angle PBA =180^{\circ}- \angle PCB$$ Let $D$ be the intersection of line $PB$ and $O$(different from $B$), and $Q$ the intersection of the tangent line of $O$ passing through $A$ and line $CD$. Show that $CQ : AB=AQ^2:AD^2$.

2009 Bosnia and Herzegovina Junior BMO TST, 4

Tags: combinatorics , circle , numbers

On circle there are $2009$ positive integers which sum is $7036$. Show that it is possible to find two pairs of neighboring numbers such that sum of both pairs is greater or equal to $8$

Denmark (Mohr) - geometry, 2003.4

Tags: geometry , circle , max

Georg and his mother love pizza. They buy a pizza shaped as an equilateral triangle. Georg demands to be allowed to divide the pizza by a straight cut and then make the first choice. The mother accepts this reluctantly, but she wants to choose a point of the pizza through which the cut must pass. Determine the largest fraction of the pizza which the mother is certain to get by this procedure.

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.

2015 India Regional MathematicaI Olympiad, 4

Tags: combinatorics , combinatorial geometry , circle

Suppose $40$ 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?

2018 Oral Moscow Geometry Olympiad, 3

Tags: orthocenter , fixed , circle , geometry

A circle is fixed, point $A$ is on it and point $K$ outside the circle. The secant passing through $K$ intersects circle at points $P$ and $Q$. Prove that the orthocenters of the triangle $APQ$ lie on a fixed circle.

2009 Abels Math Contest (Norwegian MO) Final, 3b

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

Show for any positive integer $n$ that there exists a circle in the plane such that there are exactly $n$ grid points within the circle. (A grid point is a point having integer coordinates.)

1984 IMO Longlists, 50

Tags: geometry , convex quadrilateral , circle , parallel , diameter , IMO , IMO 1984

Let $ABCD$ be a convex quadrilateral with the line $CD$ being tangent to the circle on diameter $AB$. Prove that the line $AB$ is tangent to the circle on diameter $CD$ if and only if the lines $BC$ and $AD$ are parallel.

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.

1996 Rioplatense Mathematical Olympiad, Level 3, 4

Tags: geometry , circle , Locus

Let $S$ be the circle of center $O$ and radius $R$, and let $A, A'$ be two diametrically opposite points in $S$. Let $P$ be the midpoint of $OA'$ and $\ell$ a line passing through $P$, different from $AA '$ and from the perpendicular on $AA '$. Let $B$ and $C$ be the intersection points of $\ell$ with $S$ and let $M$ be the midpoint of $BC$. a) Let $H$ be the foot of the altitude from $A$ in the triangle $ABC$. Let $D$ be the intersection point of the line $A'M$ with $AH$. Determine the locus of point $D$ while $\ell$ varies . b) Line $AM$ intersects $OD$ at $I$. Prove that $2 OI = ID$ and determine the locus of point $I$ while $\ell$ varies .

2005 Argentina National Olympiad, 5

Tags: ratio , geometry , Tangents , circle

Let $AM$ and $AN$ be the lines tangent to a circle $\Gamma$ drawn from a point $A$ $(M$ and $N$ belong to the circle). A line through $A$ cuts $\Gamma$ at $B$ and $C$ with $B$ between $A$ and $C$, and $\frac{AB}{BC} =\frac23$. If $P$ is the intersection point of $AB$ and $MN$, calculate $\frac{AP}{CP}$.

2016 Bosnia and Herzegovina Team Selection Test, 1

Tags: geometry , Tangents , circle , parallel , collinear

Let $ABCD$ be a quadrilateral inscribed in circle $k$. Lines $AB$ and $CD$ intersect at point $E$ such that $AB=BE$. Let $F$ be the intersection point of tangents on circle $k$ in points $B$ and $D$, respectively. If the lines $AB$ and $DF$ are parallel, prove that $A$, $C$ and $F$ are collinear.

2011 Kyiv Mathematical Festival, 5

Tags: combinatorial geometry , circle , geometry

Pete claims that he can draw $4$ segments of length $1$ and a circle of radius less than $\sqrt3 /3 $ on a piece of paper, such that all segments would lie inside the circle and there would be no line that intersects each of $4$ segments. Is Pete right?