Olympiad problems

1982 IMO Longlists, 41

Tags: symmetry , geometry , rectangle , IMO Longlist , circle , convex polygon , IMO Shortlist

A convex, closed figure lies inside a given circle. The figure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex figure are perpendicular). Prove that the center of the circle is a center of symmetry of the figure.

2006 Sharygin Geometry Olympiad, 6

Tags: geometry , construction , circle , tangent

a) Given a segment $AB$ with a point $C$ inside it, which is the chord of a circle of radius $R$. Inscribe in the formed segment a circle tangent to point $C$ and to the circle of radius $R$. b) Given a segment $AB$ with a point $C$ inside it, which is the point of tangency of a circle of radius $r$. Draw through $A$ and $B$ a circle tangent to a circle of radius $r$.

2013 Sharygin Geometry Olympiad, 8

Tags: geometry , distance , circle

Three cyclists ride along a circular road with radius $1$ km counterclockwise. Their velocities are constant and different. Does there necessarily exist (in a sufficiently long time) a moment when all the three distances between cyclists are greater than $1$ km? by V. Protasov

1970 Putnam, A5

Tags: Putnam , circle , ellipsoid

Determine the radius of the largest circle which can lie on the ellipsoid $$\frac{x^2 }{a^2 } +\frac{ y^2 }{b^2 } +\frac{z^2 }{c^2 }=1 \;\;\;\; (a>b>c).$$

1995 Chile National Olympiad, 2

Tags: geometry , arc , areas , circle

In a circle of radius $1$, six arcs of radius $1$ are drawn, which cut the circle as in the figure. Determine the black area. [img]https://cdn.artofproblemsolving.com/attachments/8/9/0323935be8406ea0c452b3c8417a8148c977e3.jpg[/img]

2018 EGMO, 1

Tags: geometry , EGMO , Triangle , circle , EGMO 2018

Let $ABC$ be a triangle with $CA=CB$ and $\angle{ACB}=120^\circ$, and let $M$ be the midpoint of $AB$. Let $P$ be a variable point of the circumcircle of $ABC$, and let $Q$ be the point on the segment $CP$ such that $QP = 2QC$. It is given that the line through $P$ and perpendicular to $AB$ intersects the line $MQ$ at a unique point $N$. Prove that there exists a fixed circle such that $N$ lies on this circle for all possible positions of $P$.

2002 Abels Math Contest (Norwegian MO), 3a

Tags: geometry , Tangents , circle

A circle with center in $O$ is given. Two parallel tangents tangent to the circle at points $M$ and $N$. Another tangent intersects the first two tangents at points $K$ and $L$. Show that the circle having the line segment $KL$ as diameter passes through $O$.

1986 Tournament Of Towns, (123) 5

Tags: geometry , Locus , orthocenter , circle

Find the locus of the orthocentres (i.e. the point where three altitudes meet) of the triangles inscribed in a given circle . (A. Andjans, Riga)

1969 IMO Shortlist, 3

Tags: radical axis , geometry , circle , construction , tangent , IMO Shortlist , IMO Longlist

$(BEL 3)$ Construct the circle that is tangent to three given circles.

1956 Moscow Mathematical Olympiad, 324

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

a) What is the least number of points that can be chosen on a circle of length $1956$, so that for each of these points there is exactly one chosen point at distance $1$, and exactly one chosen point at distance $2$ (distances are measured along the circle)? b) On a circle of length $15$ there are selected $n$ points such that for each of them there is exactly one selected point at distance $1$ from it, and exactly one is selected point at distance $2$ from it. (All distances are measured along the circle.) Prove that $n$ is divisible by $10$.

2010 All-Russian Olympiad Regional Round, 10.6

Tags: ratio , geometry , circle

The tangent lines to the circle $\omega$ at points $B$ and $D$ intersect at point $P$. The line passing through $P$ cuts out from circle chord $AC$. Through an arbitrary point on the segment $AC$ a straight line parallel to $BD$ is drawn. Prove that it divides the lengths of polygonal $ABC$ and $ADC$ in the same ratio. [hide=last sentence was in Russian: ]Докажите, что она делит длины ломаных ABC и ADC в одинаковых отношениях. [/hide]

2024 Middle European Mathematical Olympiad, 3

Tags: geometry , Fixed point , circle , circumcircle , Angle Chasing

Let $ABC$ be an acute scalene triangle. Choose a circle $\omega$ passing through $B$ and $C$ which intersects the segments $AB$ and $AC$ at the interior points $D$ and $E$, respectively. The lines $BE$ and $CD$ intersects at $F$. Let $G$ be a point on the circumcircle of $ABF$ such that $GB$ is tangent to $\omega$ and let $H$ be a point on the circumcircle of $ACF$ such that $HC$ is tangent to $\omega$. Prove that there exists a point $T\neq A$, independent of the choice of $\omega$, such that the circumcircle of triangle $AGH$ passes through $T$.

1977 All Soviet Union Mathematical Olympiad, 238

Tags: circle , Coloring , combinatorics , combinatorial geometry

Several black and white checkers (tokens?) are standing along the circumference. Two men remove checkers in turn. The first removes all the black ones that had at least one white neighbour, and the second -- all the white ones that had at least one black neighbour. They stop when all the checkers are of the same colour. a) Let there be $40$ checkers initially. Is it possible that after two moves of each man there will remain only one (checker)? b) Let there be $1000$ checkers initially. What is the minimal possible number of moves to reach the position when there will remain only one (checker)?

2017 India PRMO, 26

Tags: geometry , Chords , parallel , area , circle

Let $AB$ and $CD$ be two parallel chords in a circle with radius $5$ such that the centre $O$ lies between these chords. Suppose $AB = 6, CD = 8$. Suppose further that the area of the part of the circle lying between the chords $AB$ and $CD$ is $(m\pi + n) / k$, where $m, n, k$ are positive integers with gcd$(m, n, k) = 1$. What is the value of $m + n + k$ ?

1977 IMO Longlists, 5

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.

Estonia Open Senior - geometry, 1996.2.4

Tags: geometry , areas , circle , square , trigonometry

The figure shows a square and a circle with a common center $O$, with equal areas of striped shapes. Find the value of $\cos a$. [img]https://2.bp.blogspot.com/-7uwa0H42ELg/XnmsSoPMgcI/AAAAAAAALgk/pHNBqtbsdKgMhcvIRYLm_8JRpOeIYcUeACK4BGAYYCw/s400/96%2Bestonia%2Bopen%2Bs2.4.png[/img]

2019 Yasinsky Geometry Olympiad, p1

Tags: analytic geometry , area of a triangle , circle

The circle $x^2 + y^2 = 25$ intersects the abscissa in points $A$ and $B$. Let $P$ be a point that lies on the line $x = 11$, $C$ is the intersection point of this line with the $Ox$ axis, and the point $Q$ is the intersection point of $AP$ with the given circle. It turned out that the area of the triangle $AQB$ is four times smaller than the area of the triangle $APC$. Find the coordinates of $Q$.

2000 IMO, 1

Tags: geometry , circle , IMO , imo 2000 , Hi

Two circles $ G_1$ and $ G_2$ intersect at two points $ M$ and $ N$. Let $ AB$ be the line tangent to these circles at $ A$ and $ B$, respectively, so that $ M$ lies closer to $ AB$ than $ N$. Let $ CD$ be the line parallel to $ AB$ and passing through the point $ M$, with $ C$ on $ G_1$ and $ D$ on $ G_2$. Lines $ AC$ and $ BD$ meet at $ E$; lines $ AN$ and $ CD$ meet at $ P$; lines $ BN$ and $ CD$ meet at $ Q$. Show that $ EP \equal{} EQ$.

2006 Sharygin Geometry Olympiad, 9.2

Tags: Euler Circle , Nine Point Circle , geometry , fixed , circle

Given a circle, point $A$ on it and point $M$ inside it. We consider the chords $BC$ passing through $M$. Prove that the circles passing through the midpoints of the sides of all the triangles $ABC$ are tangent to a fixed circle.

2014 Indonesia MO Shortlist, C6

Tags: combinatorics , circle , arc

Determine all natural numbers $n$ so that numbers $1, 2,... , n$ can be placed on the circumference of a circle and for each natural number $s$ with $1\le s \le \frac12n(n+1)$ , there is a circular arc which has the sum of all numbers in that arc to be $s$.

2014 Switzerland - Final Round, 10

Tags: geometry , circle , bisects segment

Let $k$ be a circle with diameter $AB$. Let $C$ be a point on the straight line $AB$, so that $B$ between $A$ and $C$ lies. Let $T$ be a point on $k$ such that $CT$ is a tangent to $k$. Let $l$ be the parallel to $CT$ through $A$ and $D$ the intersection of $l$ and the perpendicular to $AB$ through $T$. Show that the line $DB$ bisects segment $CT$.

1945 Moscow Mathematical Olympiad, 105

Tags: geometry , rolls , circle , arc , Equilateral

A circle rolls along a side of an equilateral triangle. The radius of the circle is equal to the height of the triangle. Prove that the measure of the arc intercepted by the sides of the triangle on this circle is equal to $60^o$ at all times.

2019 Junior Balkan Team Selection Tests - Romania, 4

Tags: combinatorics , numbers , circle

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.

2000 Tournament Of Towns, 3

Tags: geometry , rectangle , Locus , circle , fixed

$A$ is a fixed point inside a given circle. Determine the locus of points $C$ such that $ABCD$ is a rectangle with $B$ and $D$ on the circumference of the given circle. (M Panov)

2006 Korea Junior Math Olympiad, 7

Tags: geometry , circle , tangent , perpendicular

A line through point $P$ outside of circle $O$ meets the said circle at $B,C$ ($PB < PC$). Let $PO$ meet circle $O$ at $Q,D$ (with $PQ < PD$). Let the line passing $Q$ and perpendicular to $BC$ meet circle $O$ at $A$. If $BD^2 = AD\cdot CP$, prove that $PA$ is a tangent to $O$.