Olympiad problems

2000 Croatia National Olympiad, Problem 2

Let $ABC$ be a triangle with $AB = AC$. With center in a point of the side $BC$, the circle $S$ is constructed that is tangent to the sides $AB$ and $AC$. Let $P$ and $Q$ be any points on the sides $AB$ and $AC$ respectively, such that $PQ$ is tangent to $S$. Show that $PB \cdot CQ = \left(\frac{BC}{2}\right)^2$

Geometry Mathley 2011-12, 3.4

Tags: circumcircle , circles , geometry , tangent circle

A triangle $ABC$ is inscribed in the circle $(O,R)$. A circle $(O',R')$ is internally tangent to $(O)$ at $I$ such that $R < R'$. $P$ is a point on the circle $(O)$. Rays $PA, PB, PC$ meet $(O')$ at $A_1,B_1,C_1$. Let $A_2B_2C_2$ be the triangle formed by the intersections of the line symmetric to $B_1C_1$ about $BC$, the line symmetric to $C_1A_1$ about $CA$ and the line symmetric to $A_1B_1$ about $AB$. Prove that the circumcircle of $A_2B_2C_2$ is tangent to $(O)$. Nguyễn Văn Linh

2012 Dutch BxMO/EGMO TST, 2

Tags: geometry , circles , circumcircle , concyclic

Let $\triangle ABC$ be a triangle and let $X$ be a point in the interior of the triangle. The second intersection points of the lines $XA,XB$ and $XC$ with the circumcircle of $\triangle ABC$ are $P,Q$ and $R$. Let $U$ be a point on the ray $XP$ (these are the points on the line $XP$ such that $P$ and $U$ lie on the same side of $X$). The line through $U$ parallel to $AB$ intersects $BQ$ in $V$ . The line through $U$ parallel to $AC$ intersects $CR$ in $W$. Prove that $Q, R, V$ , and $W$ lie on a circle.

2022 AMC 12/AHSME, 25

Tags: circles , geometry , excircle

A circle with integer radius $r$ is centered at $(r, r)$. Distinct line segments of length $c_i$ connect points $(0, a_i)$ to $(b_i, 0)$ for $1 \le i \le 14$ and are tangent to the circle, where $a_i$, $b_i$, and $c_i$ are all positive integers and $c_1 \le c_2 \le \cdots \le c_{14}$. What is the ratio $\frac{c_{14}}{c_1}$ for the least possible value of $r$? $\textbf{(A)} ~\frac{21}{5} \qquad\textbf{(B)} ~\frac{85}{13} \qquad\textbf{(C)} ~7 \qquad\textbf{(D)} ~\frac{39}{5} \qquad\textbf{(E)} ~17 $

2009 Sharygin Geometry Olympiad, 8

Tags: circumcircle , cyclic quadrilateral , circles , geometry

Given cyclic quadrilateral $ABCD$. Four circles each touching its diagonals and the circumcircle internally are equal. Is $ABCD$ a square? (C.Pohoata, A.Zaslavsky)

1997 Croatia National Olympiad, Problem 3

Tags: circles , geometry

A chord divides the interior of a circle $k$ into two parts. Variable circles $k_1$ and $k_2$ are inscribed in these two parts, touching the chord at the same point. Show that the ratio of the radii of circles $k_1$ and $k_2$ is constant, i.e. independent of the tangency point with the chord.

1998 Belarus Team Selection Test, 1

Tags: geometry , locus , circles , angle

Two circles $S_1$ and $S_2$ intersect at different points $P,Q$. The arc of $S_1$ lying inside $S_2$ measures $2a$ and the arc of $S_2$ lying inside $S_1$ measures $2b$. Let $T$ be any point on $S_1$. Let $R,S$ be another points of intersection of $S_2$ with $TP$ and $TQ$ respectively. Let $a+2b<\pi$ . Find the locus of the intersection points of $PS$ and $RQ$. S.Shikh

2014 IFYM, Sozopol, 2

Tags: point , circles , prime , number theory

The radius $r$ of a circle with center at the origin is an odd integer. There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers. Determine $r$.

Denmark (Mohr) - geometry, 1998.1

Tags: geometry , circles , equal circles , tangent circle

In the figure shown, the small circles have radius $1$. Calculate the area of the gray part of the figure. [img]https://1.bp.blogspot.com/-oy-WirJ6u9o/XzcFc3roVDI/AAAAAAAAMX8/qxNy5I_0RWUOxl-ZE52fnrwo0v0T7If9QCLcBGAsYHQ/s0/1998%2BMohr%2Bp1.png[/img]

2017 Turkey Team Selection Test, 3

Tags: circles , incenter , geometry

At the $ABC$ triangle the midpoints of $BC, AC, AB$ are respectively $D, E, F$ and the triangle tangent to the incircle at $G$, $H$ and $I$ in the same order.The midpoint of $AD$ is $J$. $BJ$ and $AG$ intersect at point $K$. The $C-$centered circle passing through $A$ cuts the $[CB$ ray at point $X$. The line passing through $K$ and parallel to the $BC$ and $AX$ meet at $U$. $IU$ and $BC$ intersect at the $P$ point. There is $Y$ point chosen at incircle. $PY$ is tangent to incircle at point $Y$. Prove that $D, E, F, Y$ are cyclic.

2018 Dutch BxMO TST, 1

Tags: coloring , circles , combinatorics

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.

1985 IMO Longlists, 5

Tags: geometry , construction , triangle , circles

If possible, construct an equilateral triangle whose three vertices are on three given circles.

1967 IMO Shortlist, 4

Tags: analytic geometry , geometry , geometric inequality , circles

Let $k_1$ and $k_2$ be two circles with centers $O_1$ and $O_2$ and equal radius $r$ such that $O_1O_2 = r$. Let $A$ and $B$ be two points lying on the circle $k_1$ and being symmetric to each other with respect to the line $O_1O_2$. Let $P$ be an arbitrary point on $k_2$. Prove that \[PA^2 + PB^2 \geq 2r^2.\]

2014 Peru Iberoamerican Team Selection Test, P1

Tags: circles , pyramid , geometry

Circles $C_1$ and $C_2$ intersect at different points $A$ and $B$. The straight lines tangents to $C_1$ that pass through $A$ and $B$ intersect at $T$. Let $M$ be a point on $C_1$ that is out of $C_2$. The $MT$ line intersects $C_1$ at $C$ again, the $MA$ line intersects again to $C_2$ in $K$ and the line $AC$ intersects again to the circumference $C_2$ in $L$. Prove that the $MC$ line passes through the midpoint of the $KL$ segment.

1947 Moscow Mathematical Olympiad, 127

Tags: equal circles , circles , geometry , circumcircle

Point $O$ is the intersection point of the heights of an acute triangle $\vartriangle ABC$. Prove that the three circles which pass: a) through $O, A, B$, b) through $O, B, C$, and c) through $O, C, A$, are equal

2014 Belarus Team Selection Test, 1

Tags: geometry , circles , concyclic , collinear

Circles $\Gamma_1$ and $\Gamma_2$ meet at points $X$ and $Y$. A circle $S_1$ touches internally $\Gamma_1$ at $A$ and $\Gamma_2$ externally at $B$. A circle $S_2$ touches $\Gamma_2$ internally at $C$ and $\Gamma_1$ externally at $D$. Prove that the points $A, B, C, D$ are either collinear or concyclic. (A. Voidelevich)

2008 Princeton University Math Competition, B6

Tags: circles

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]

2009 Oral Moscow Geometry Olympiad, 5

Tags: distance , circles , geometry

A treasure is buried at some point on a round island with a radius of $1$ km. On the coast of the island there is a mathematician with a device that indicates the direction to the treasure when the distance to the treasure does not exceed $500$ m. In addition, the mathematician has a map of the island, on which he can record all his movements, perform measurements and geometric constructions. The mathematician claims that he has an algorithm for how to get to the treasure after walking less than $4$ km. Could this be true? (B. Frenkin)

1997 Estonia National Olympiad, 5

Tags: geometry , rectangle , circles

Find the length of the longer side of the rectangle on the picture, if the shorter side has length $1$ and the circles touch each other and the sides of the rectangle as shown. [img]https://cdn.artofproblemsolving.com/attachments/b/8/3986683247293bd089d8e83911309308ce0c3a.png[/img]

1995 Chile National Olympiad, 2

Tags: geometry , arc , area , circles

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]

2012 Ukraine Team Selection Test, 2

Tags: geometry , circumcircle , cyclic , concyclic , circles

$E$ is the intersection point of the diagonals of the cyclic quadrilateral, $ABCD, F$ is the intersection point of the lines $AB$ and $CD, M$ is the midpoint of the side $AB$, and $N$ is the midpoint of the side $CD$. The circles circumscribed around the triangles $ABE$ and $ACN$ intersect for the second time at point $K$. Prove that the points $F, K, M$ and $N$ lie on one circle.

2000 Kazakhstan National Olympiad, 2

Tags: geometry , equal segments , circles

Given a circle centered at $ O $ and two points $ A $ and $ B $ lying on it. $ A $ and $ B $ do not form a diameter. The point $ C $ is chosen on the circle so that the line $ AC $ divides the segment $ OB $ in half. Let lines $ AB $ and $ OC $ intersect at $ D $, and let lines $ BC $ and $ AO $ intersect at $ F $. Prove that $ AF = CD $.

1984 IMO Longlists, 50

Tags: convex quadrilateral , parallel , diameter , circles , geometry

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.

2015 Dutch IMO TST, 4

Tags: geometry , perpendicular , parallel , circles

Let $\Gamma_1$ and $\Gamma_2$ be circles - with respective centres $O_1$ and $O_2$ - that intersect each other in $A$ and $B$. The line $O_1A$ intersects $\Gamma_2$ in $A$ and $C$ and the line $O_2A$ intersects $\Gamma_1$ in $A$ and $D$. The line through $B$ parallel to $AD$ intersects $\Gamma_1$ in $B$ and $E$. Suppose that $O_1A$ is parallel to $DE$. Show that $CD$ is perpendicular to $O_2C$.

1982 All Soviet Union Mathematical Olympiad, 327

Tags: geometry , circles , area

Given two points $M$ and $K$ on the circumference with radius $r_1$ and centre $O_1$. The circumference with radius $r_2$ and centre $O_2$ is inscribed in $\angle MO_1K$ . Find the area of quadrangle $MO_1KO_2$ .