Olympiad problems

2000 Switzerland Team Selection Test, 9

Two given circles $k_1$ and $k_2$ intersect at points $P$ and $Q$. Construct a segment $AB$ through $P$ with the endpoints at $k_1$ and $k_2$ for which $AP \cdot PB$ is maximal.

2004 Singapore MO Open, 3

Tags: perpendicular , intersecting circles , concyclic , circles , geometry

Let $AD$ be the common chord of two circles $\Gamma_1$ and $\Gamma_2$. A line through $D$ intersects $\Gamma_1$ at $B$ and $\Gamma_2$ at $C$. Let $E$ be a point on the segment $AD$, different from $A$ and $D$. The line $CE$ intersect $\Gamma_1$ at $P$ and $Q$. The line $BE$ intersects $\Gamma_2$ at $M$ and $N$. (i) Prove that $P,Q,M,N$ lie on the circumference of a circle $\Gamma_3$. (ii) If the centre of $\Gamma_3$ is $O$, prove that $OD$ is perpendicular to $BC$.

1995 Abels Math Contest (Norwegian MO), 2b

Tags: circles , perpendicular bisector , intersecting circles , equal circles

Two circles of the same radii intersect in two distinct points $P$ and $Q$. A line passing through $P$, not touching any of the circles, intersects the circles again at $A$ and $B$. Prove that $Q$ lies on the perpendicular bisector of $AB$.

1999 Switzerland Team Selection Test, 1

Tags: tangent , geometry , parallel , intersecting circles

Two circles intersect at points $M$ and $N$. Let $A$ be a point on the first circle, distinct from $M,N$. The lines $AM$ and $AN$ meet the second circle again at $B$ and $C$, respectively. Prove that the tangent to the first circle at $A$ is parallel to $BC$.

1995 Grosman Memorial Mathematical Olympiad, 4

Tags: geometry , circles , locus , intersecting circles

Two given circles $\alpha$ and $\beta$ intersect each other at two points. Find the locus of the centers of all circles that are orthogonal to both $\alpha$ and $\beta$.

2004 Estonia National Olympiad, 5

Tags: geometry , circles , intersecting circles , center , tangent circle

Three different circles of equal radii intersect in point $Q$. The circle $C$ touches all of them. Prove that $Q$ is the center of $C$.