\documentclass[12pt]{amsart}
\usepackage{graphics}
\usepackage[all]{xy}
\begin{document}
\begin{enumerate}
\item
Let $\Gamma$ be a circle with center $O$ and radius $r$ and let $A$ be a point in the exterior of $\Gamma$.
Let $M$ be a point on $\Gamma$ and let $N$ be the point on $\Gamma$ such that $MN$ is a diameter.
Determine the locus of the centers of the circles which pass through $A$, $M$, and $N$ as one varies $M$.
\item
Let $\triangle ABC$ be an obtuse angled triangle and let $A'$, $B'$, and $C'$ (respectively) be the points of
intersection of the interior angle bisectors of angles $A$, $B$, and $C$ (respectively)
with the opposite sides of the triangle. Now let:
\begin{itemize}
\item $A''$ be the intersection of $BC$ with the perpendicular
bisector of $AA'$;
\item $B''$ be the intersection of $AC$ with the perpendicular
bisector of $BB'$;
\item $C''$ be the intersection of $AB$ with the perpendicular
bisector of $CC'$.
\end{itemize}
Show that $A''$, $B''$, and $C''$ are collinear.
\item
Let $O$ be the circumcenter of an acute angled triangle $ABC$ and $A_1$ a point on the arc
$BC$ which is part of the circumcircle of the triangle $ABC$.
Let $A_2$ and $A_3$ be points on the sides $AB$ and $AC$ respectively, such that $\angle BA_1A_2=\angle OAC$ and
$\angle CA_1A_3=\angle OAB$. Show that the line segment $A_2A_3$ passes through the orthocenter of the triangle
$ABC$.
\item
Given a set $S$ of points in the plane, we call a circle in the plane a 4-circle if it passes through at least four points of $S$. What is the maximum number of 4-circles that could be determined
by a set of 7 points?
\item
We assign a real number between 0 and 1 to every point of the plane with integer coordinates.
This is done in such a way that the number assigned to a given point is equal to the
arithmetic mean of the numbers assigned to the four points that have distance one to the given point
(the points directly above, below, to the left, and to the right of the given point).
Show that all the numbers are equal.
\item
Determine the smallest real number $r$ such that it is possible to cover an equilateral
triangle with side length 1 by six circles with radius $r$.
\item
Let $ABC$ be an equilateral triangle and $P$ an interior point such that $\angle APC=120^\circ$.
Let $M$ be the intersection of $CP$ with $AB$ and $N$ be the intersection of $AP$ with $BC$.
Find the locus of the circumcenter of the triangle $MBN$ as we vary $P$.
\item
Given a circle $\Gamma$, consider a quadrilateral $ABCD$ with its four sides tangent to $\Gamma$.
Let $AD$ be tangent to $\Gamma$ at $P$ and $CD$ be tangent to $\Gamma$ at $Q$.
Let $X$ and $Y$ be the points where the segment $BD$ intersects $\Gamma$, and let $M$ be the
midpoint of $XY$.
Show that $\angle AMP=\angle CMQ$.
\item
Let $M$ and $N$ be points on the sides $AC$ and $BC$ (respectively) of a triangle $ABC$,
and let $P$ be a point on the line segment $MN$. Show that at least one of the triangles
$AMP$ and $BNP$ has an area which is less than or equal to $\frac{1}{8}$ of the area of the
triangle $ABC$.
\item
Let $ABCD$ be a convex quadrilateral. The extensions of $AB$ and $CD$ intersect in $E$,
and the extensions of $AD$ and $BC$ intersect in $F$. The angle bisectors of $\angle A$ and $\angle C$
intersect in $P$, and the angle bisector of $\angle B$ and $\angle D$ intersect in $Q$.
The angle bisectors of the exterior angles at $E$ (for triangle $ADE$) and $F$ (for triangle $ABF$)
intersect in $R$. Show that $P$, $Q$, and $R$ are collinear.
\end{enumerate}
\end{document}