Olympiad problems

1985 IMO Shortlist, 20

A circle whose center is on the side $ED$ of the cyclic quadrilateral $BCDE$ touches the other three sides. Prove that $EB+CD = ED.$

1966 All Russian Mathematical Olympiad, 078

Tags: convex polygon , circle , geometry

Prove that you can always pose a circle of radius $S/P$ inside a convex polygon with the perimeter $P$ and area $S$.

1967 IMO Shortlist, 6

Tags: combinatorics , geometry , counting , circle , IMO Shortlist , IMO Longlist

On the circle with center 0 and radius 1 the point $A_0$ is fixed and points $A_1, A_2, \ldots, A_{999}, A_{1000}$ are distributed in such a way that the angle $\angle A_00A_k = k$ (in radians). Cut the circle at points $A_0, A_1, \ldots, A_{1000}.$ How many arcs with different lengths are obtained. ?

Kvant 2019, M2553

Tags: geometry , orthocenter , circle , Incenters , angle bisector , Kvant

A circle centred at $I$ is tangent to the sides $BC, CA$, and $AB$ of an acute-angled triangle $ABC$ at $A_1, B_1$, and $C_1$, respectively. Let $K$ and $L$ be the incenters of the quadrilaterals $AB_1IC_1$ and $BA_1IC_1$, respectively. Let $CH$ be an altitude of triangle $ABC$. Let the internal angle bisectors of angles $AHC$ and $BHC$ meet the lines $A_1C_1$ and $B_1C_1$ at $P$ and $Q$, respectively. Prove that $Q$ is the orthocenter of the triangle $KLP$. Kolmogorov Cup 2018, Major League, Day 3, Problem 1; A. Zaslavsky

2008 Princeton University Math Competition, B6

Tags: circle

Circles $A, B$, and $C$ each have radius $r$, and their centers are the vertices of an equilateral triangle of side length $6r$. Two lines are drawn, one tangent to $A$ and $C$ and one tangent to $B$ and $C$, such that $A$ is on the opposite side of each line from $B$ and $C$. Find the sine of the angle between the two lines. [img]http://4.bp.blogspot.com/-IZv8q-3NYZg/XXmrroy2PnI/AAAAAAAAKxg/jSOcOOQ8Kyw0EwHUifXJ1jOd2ENAo1FfACK4BGAYYCw/s200/2008%2Bpumac%2Bb6.png[/img]

1961 All-Soviet Union Olympiad, 2

Tags: geometry , rectangle , circle , combinatorial geometry

Consider $120$ unit squares arbitrarily situated in a $20\times 25$ rectangle. Prove that one can place a circle with unit diameter in the rectangle without intersecting any of the squares.

Champions Tournament Seniors - geometry, 2017.4

Tags: equal segments , geometry , circle , angle bisector , Champions Tournament

Let $AD$ be the bisector of triangle $ABC$. Circle $\omega$ passes through the vertex $A$ and touches the side $BC$ at point $D$. This circle intersects the sides $AC$ and $AB$ for the second time at points $M$ and $N$ respectively. Lines $BM$ and $CN$ intersect the circle for the second time $\omega$ at points $P$ and $Q$, respectively. Lines $AP$ and $AQ$ intersect side $BC$ at points $K$ and $L$, respectively. Prove that $KL=\frac12 BC$

2014 Hanoi Open Mathematics Competitions, 14

Tags: geometry , Fixed point , circle

Let $\omega$ be a circle with centre $O$, and let $\ell$ be a line that does not intersect $\omega$. Let $P$ be an arbitrary point on $\ell$. Let $A,B$ denote the tangent points of the tangent lines from $P$. Prove that $AB$ passes through a point being independent of choosing $P$.

2017 Germany, Landesrunde - Grade 11/12, 2

Tags: ratio , invariant , geometry , circle , Germany

Three circles $k_1,k_2$ and $k_3$ go through the points $A$ and $B$. A secant through $A$ intersects the circles $k_1,k_2$ and $k_3$ again in the points $C,D$ resp. $E$. Prove that the ratio $|CD|:|DE|$ does not depend on the choice of the secant.

2017 Oral Moscow Geometry Olympiad, 1

Tags: geometry , angle , Squares , circle

One square is inscribed in a circle, and another square is circumscribed around the same circle so that its vertices lie on the extensions of the sides of the first (see figure). Find the angle between the sides of these squares. [img]https://3.bp.blogspot.com/-8eLBgJF9CoA/XTodHmW87BI/AAAAAAAAKY0/xsHTx71XneIZ8JTn0iDMHupCanx-7u4vgCK4BGAYYCw/s400/sharygin%2Boral%2B2017%2B10-11%2Bp1.png[/img]

2022 3rd Memorial "Aleksandar Blazhevski-Cane", P6

Tags: combinatorics , circle , central symmetry , perfect matching , symmetry

For any integer $n\geq1$, we consider a set $P_{2n}$ of $2n$ points placed equidistantly on a circle. A [i]perfect matching[/i] on this point set is comprised of $n$ (straight-line) segments whose endpoints constitute $P_{2n}$. Let $\mathcal{M}_{n}$ denote the set of all non-crossing perfect matchings on $P_{2n}$. A perfect matching $M\in \mathcal{M}_{n}$ is said to be [i]centrally symmetric[/i], if it is invariant under point reflection at the circle center. Determine, as a function of $n$, the number of centrally symmetric perfect matchings within $\mathcal{M}_{n}$. [i]Proposed by Mirko Petrusevski[/i]

1986 Tournament Of Towns, (120) 2

Tags: geometry , perimeter , Sum , circle , square

Square $ABCD$ and circle $O$ intersect in eight points, forming four curvilinear triangles, $AEF , BGH , CIJ$ and $DKL$ ($EF , GH, IJ$ and $KL$ are arcs of the circle) . Prove that (a) The sum of lengths of $EF$ and $IJ$ equals the sum of the lengths of $GH$ and $KL$. (b) The sum of the perimeters of curvilinear triangles $AEF$ and $CIJ$ equals the sum of the perimeters of the curvilinear triangles $BGH$ and $DKL$. ( V . V . Proizvolov , Moscow)

2011 Sharygin Geometry Olympiad, 4

Tags: circle , geometry , regular polygon , hexagon , inscribed , Chords , combinatorics

Given the circle of radius $1$ and several its chords with the sum of lengths $1$. Prove that one can be inscribe a regular hexagon into that circle so that its sides don’t intersect those chords.

2017 Thailand Mathematical Olympiad, 10

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

A lattice point is defined as a point on the plane with integer coordinates. Show that for all positive integers $n$, there is a circle on the plane with exactly n lattice points in its interior (not including its boundary).

2012 Tournament of Towns, 4

Tags: parallelogram , bisects segment , geometry , circle

A circle touches sides $AB, BC, CD$ of a parallelogram $ABCD$ at points $K, L, M$ respectively. Prove that the line $KL$ bisects the height of the parallelogram drawn from the vertex $C$ to $AB$.

1984 All Soviet Union Mathematical Olympiad, 384

Tags: areas , polygon , perimeter , circle , discs , geometry

The centre of the coin with radius $r$ is moved along some polygon with the perimeter $P$, that is circumscribed around the circle with radius $R$ ($R>r$). Find the coin trace area (a sort of polygon ring).

2015 Switzerland - Final Round, 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 $k$.

2018 Dutch BxMO TST, 1

Tags: combinatorics , circle , Coloring

We have $1000$ balls in $40$ different colours, $25$ balls of each colour. Determine the smallest $n$ for which the following holds: if you place the $1000$ balls in a circle, in any arbitrary way, then there are always $n$ adjacent balls which have at least $20$ different colours.

2017 Finnish National High School Mathematics Comp, 5

Tags: circle , Circumcenter , diameter , geometry

Let $A$ and $B$ be two arbitrary points on the circumference of the circle such that $AB$ is not the diameter of the circle. The tangents to the circle drawn at points $A$ and $B$ meet at $T$. Next, choose the diameter $XY$ so that the segments $AX$ and $BY$ intersect. Let this be the intersection of $Q$. Prove that the points $A, B$, and $Q$ lie on a circle with center $T$.

1966 IMO Longlists, 16

Tags: geometry , Locus problems , Locus , circle , IMO Shortlist , IMO Longlist

We are given a circle $K$ with center $S$ and radius $1$ and a square $Q$ with center $M$ and side $2$. Let $XY$ be the hypotenuse of an isosceles right triangle $XY Z$. Describe the locus of points $Z$ as $X$ varies along $K$ and $Y$ varies along the boundary of $Q.$

2016 Hanoi Open Mathematics Competitions, 12

Tags: geometry , fixed , circle

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 $A$).

1977 IMO Shortlist, 4

Tags: geometry , circle , Convexity

Describe all closed bounded figures $\Phi$ in the plane any two points of which are connectable by a semicircle lying in $\Phi$.

2010 Sharygin Geometry Olympiad, 2

Tags: geometry , Locus , circle

Two points $A$ and $B$ are given. Find the locus of points $C$ such that triangle $ABC$ can be covered by a circle with radius $1$. (Arseny Akopyan)

1998 Portugal MO, 5

Tags: geometry , chord , arc midpoint , circle

Let $F$ be the midpoint of circle arc $AB$, and let $M$ be a point on the arc such that $AM <MB$. The perpendicular drawn from point $F$ on $AM$ intersects $AM$ at point $T$. Show that $T$ bisects the broken line $AMB$, that is $AT =TM+MB$. KöMaL Gy. 2404. (March 1987), Archimedes of Syracuse

2008 Estonia Team Selection Test, 5

Tags: tangent , midpoints , circle , geometry

Points $A$ and $B$ are fixed on a circle $c_1$. Circle $c_2$, whose centre lies on $c_1$, touches line $AB$ at $B$. Another line through $A$ intersects $c_2$ at points $D$ and $E$, where $D$ lies between $A$ and $E$. Line $BD$ intersects $c_1$ again at $F$. Prove that line $EB$ is tangent to $c_1$ if and only if $D$ is the midpoint of the segment $BF$.