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{\footnotesize December 2004}

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\begin{centre}
  \bf \Large
 Plane Geometry (2) (S)
\end{centre}

\begin{enumerate}

\item Let $ABCD$ be a rhombus and P be a point on its side $BC$.
The circle passing through $A$, $B$, and $P$ intersects $BD$ once
more at the point $Q$ and the circle passing through $C$, $P$, and
$Q$ intersects $BD$ once more at the point $R$. Prove that $A$,
$R$, and $P$ lie on the one straight line. [Tournament of the
Towns 1990]


\item The chord $MN$ on the circle is fixed. For every diameter
$AB$ of the circle consider the intersection point $C$ of the
lines $AM$ and $BN$ and construct the line $\ell$ passing through
$C$ perpendicularly to $AB$. Prove that all the lines $\ell$ pass
through a fixed point. [Tournament of the Towns 1991]


\item The points $P$ and $Q$ lie on a semi-circle with diameter
$UV$. $UP$ and $VQ$ intersect at the point $S$, while the tangents
to the semi-circle at $P$ and $Q$ intersect at $R$. Prove that $RS
\bot UV$.

\item The incentre of the triangle $\triangle ABC$ is $K$. The
midpoint of $AB$ is $C_1$ and that of $AC$ is $B_1$. The lines
$C_1K$ and $AC$ meet at $B_2$, the lines $B_1K$ and $AB$ meet at
$C_2$. If the areas of the triangles $\triangle AB_2C_2$ and
$\triangle ABC$ are equal, what is the measure of $\angle CAB$?
[1990 IMO Shortlist - HUN 3]


\item Let $OA$ and $OB$ be perpendicular rays in the circle
$\mathcal{C}$ (with centre $O$). Circles $\mathcal{C}_1$ and
$\mathcal{C}_2$ are internally tangent to $\mathcal{C}$ in $A$ and
$B$, and a third circle $\mathcal{C}_3$ is tangent externally to
$\mathcal{C}_1$ and $\mathcal{C}_2$ in $S$ and $T$, and internally
in $M$ to $\mathcal{C}$. Find the measure of $\angle SMT$. [1996
Romanian Mathematical Olympiad]


\item If $ABCDEF$ is a convex hexagon with $AB = BC$, $CD = DE$,
$EF = FA$, prove that the altitudes (produced) of $\triangle BCD$,
$\triangle DEF$ and $\triangle FAB$, emanating from vertices $C$,
$E$, $A$, concur. [Polish and Austrian Olympiads 1981--1995]


\item Let $ABC$ be an acute triangle with altitudes $BD$ and $CE$.
Points $F$ and $G$ are the feet of the perpendiculars $BF$ and
$CG$ to line $DE$. Prove that $EF = DG$. [Polish and Austrian
Olympiads 1981--1995]


\item Points $D$, $E$, $F$ are chosen on the sides $AB$, $BC$,
$AC$ of a triangle $ABC$, so that $DE = BE$ and $FE = CE$. Prove
that the centre of the circle circumscribed around triangle $ADF$
lies on the bisector of $\angle DEF$. [USSR Olympiad 1989]


\item Two common tangents of two intersecting circles meet at a
point $A$. Let $B$ be a point of intersection of the two circles,
and $C$ and $D$ be the points at which one of the tangents touches
the circles. Prove that the line $AB$ is tangent to the circle
passing through $B$, $C$, and $D$. [USSR Olympiad 1990]


\item On the side $AB$ of a convex quadrilateral $ABCD$ a point
$E$, different from the vertices, is chosen. The segments $AC$ and
$DE$ intersect at a point $F$. Prove that the circles
circumscribed about $\triangle ABC$, $\triangle CDF$, and
$\triangle BDE$ have a common point. [USSR Olympiad 1990]

\item Let triangle $ABC$ have orthocentre $H$, and let $P$ be a
point on its circumcircle. Let $E$ be the foot of the altitude
$BH$, let $PAQB$ and $PARC$ be parallelograms, and let $AQ$ meet
$HR$ in $X$. Prove that $EX$ is parallel to $AP$.

\item (IMO 1996 Q2) Let $P$ be a point inside $\triangle ABC$ such
that $\angle APB-\angle C=\angle APC-\angle B$. Let $D$, $E$ be
the incentres of $\triangle APB$, $\triangle APC$ respectively.
Show that $AP$, $BD$ and $CE$ meet in a point.

\item Let $ABC$ be an acute-angled triangle with $BC>CA$. Let $O$
be its circumcentre, $H$ its orthocentre, and $F$ the foot of its
altitude $CH$. Let the perpendicular to $OF$ at $F$ meet the side
$CA$ at $P$. Prove that $\angle FHP=\angle BAC$.

\item In an acute triangle $ABC$, $AC>BC$, $M$ is the midpoint of
$AB$. Let $AP$ be the altitude from $A$, $BQ$ be the altitude from
$B$, $AP$ and $BQ$ meet in $H$, and let the lines $AB$ and $PQ$
meet at $R$. Prove that the  two lines $RH$ and $CM$ are
perpendicular.

\item (IMO 1990 Q1) Chords $AB$ and $CD$ of a circle intersect at
a point $E$ inside the circle. Let $M$ be an interior point of the
segment $EB$. The tangent line at $E$ to the circle through $D$,
$E$ and $M$ intersects the lines $BC$ and $AC$ at $F$ and $G$,
respectively. If $\frac{AM}{AB}=t$, find $\frac{EG}{EF}$ in terms
of $t$.

\item (Czech-Slovak 1999) An acute angled triangle $ABC$ is given
with altitudes $AD$, $BE$, $CF$. Suppose that the lines $BC$ and
$EF$ have a point $P$ in common and that the line through $D$
parallel to $EF$ intersects the line $AC$ at a point $Q$ and the
line $AB$ at a point $R$. Prove that the circumcircle of the
triangle $PQR$ passes through the midpoint of the side $BC$.

\item (IMO 1999 Q5) The circles $\Gamma_1$ and $\Gamma_2$ lie
inside circle $\Gamma$, and are tangent to it at $M$ and $N$
respectively. It is given that $\Gamma_1$ passes through the
centre of $\Gamma_2$. The common chord of $\Gamma_1$ and
$\Gamma_2$, when extended, meets $\Gamma$ at $A$ and $B$. The
lines $MA$ and $MB$ meet $\Gamma_1$ again at $C$ and $D$. Prove
that the line $CD$ is tangent to $\Gamma_2$.

\item (Czech-Slovak 1999) Find all positive numbers $k$ for which
the following assertion holds: Among all triangles $ABC$ with
$|AB|=5$ and $|AC|:|BC|=k$, the one with the largest area is the
isosceles one.

\item Consider an acute angled triangle $ABC$ such that $AC>BC$,
and let $M$ be the midpoint of $AB$ and let $CD$, $AP$ and $BQ$ be
the altitudes of the triangle. Let $R$ be the intersection point
of $AB$ and $PQ$. Prove that $MP$ is tangent to the circumcircle
of $DRP$.

\end{enumerate}


\end{document}
