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\begin{center}
${\bf 36}^{\mbox{\bf th}}$
{\bf International Mathematical Olympiad} \\[.1in]
{\bf First Day - Toronto - July 19, 1995} \\
{\bf Time Limit: 4}${{\bf 1}\over{\bf 2}}$ {\bf
hours}
\end{center}
\begin{enumerate}
\item
Let $A,B,C, D$ be four distinct points on a line, in that order. The
circles with diameters $AC$ and $BD$ intersect at $X$ and $Y$.
The line $XY$ meets $BC$ at $Z$. Let $P$ be a point on the line
$XY$ other than $Z$. The line $CP$ intersects the circle with diameter
$AC$ at $C$ and $M$, and the line $BP$ intersects the circle
with diameter $BD$ at $B$ and $N$. Prove that the lines $AM,
DN, XY$ are concurrent.
\item
Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that
\[
\frac{1}{a^3(b+c)} + \frac{1}{b^3(c+a)} + \frac{1}{c^3(a+b)} \geq \frac{3}{2}.
\]
\item
Determine all integers $n > 3$ for which there exist $n$ points $A_1,
\dots, A_n$ in the plane, no three collinear, and real numbers $r_1, \dots,
r_n$ such that for $1 \leq i