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\begin{document}
\begin{center}
{\bf \Large Singapore International Mathematical Olympiad}

\bigskip
{\bf \Large National Team Selection Test 1}
\end{center}

\bigskip
{\bf Time allowed: 4.5 hours}\hfill{\bf 29 December 1995}

\medskip
{\bf No calculator is allowed}

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\item Let $P$ be a point on the side $AB$ of a square $ABCD$ 
and $Q$ a point on the side $BC$.  Let $H$ be the foot of the 
perpendicular from $B$ to $PC$.  Suppose that $BP = BQ$.
Prove that $QH$ is perpendicular to $HD$.

\item For each positive integer $k$, prove that there is 
a perfect square of the form $n2^{k}-7$, where $n$ is a positive integer.  
 
 
\item Let $S=\{0,1,2,\dots,1994\}$. Let $a$ and $b$ be two positive numbers 
in $S$ which are relatively prime. Prove that the elements of $S$ can be 
arranged into a sequence $s_{1},s_{2},s_{3},\dots,s_{1995}$ such that 
$s_{i+1}-s_{i} \equiv \pm{a} \ \mbox{or} \pm{b}$ \ (mod $1995$) 
for $i = 1,2,\dots,1994$.

\end{list}

\newpage
\begin{center}
{\bf \Large  Singapore International Mathematical Olympiad}

\bigskip
{\bf \Large  National Team Selection Test 2}
\end{center}

\bigskip
{\bf Time allowed: 4.5 hours}\hfill{\bf 20 January 1996}

\medskip
{\bf No calculator is allowed}

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\item Let $C,B,E$ be three points on a straight line $l$ in that order. 
Suppose that $A$ and $D$ are two points on the same side of $l$ such that 

(i)\  $\angle ACE = \angle CDE = 90^{o}$ and 

(ii) CA = CB = CD.
 
Let $F$ be the point of intersection of 
the segment $AB$ and the circumcircle of $\triangle ADC$. 
Prove that $F$ is the incentre of $\triangle CDE$.

\item Prove that there is a function $f$ from the set of all natural numbers 
to itself such that for any natural number $n$, $f(f(n)) = n^{2}$.

\item Let $S$ be a sequence $n_{1},n_{2},\dots,n_{1995}$ of positive 
integers such that $n_{1} + \cdots + n_{1995} = m < 3990$. Prove that 
for each integer $q$ with $1\leq q\leq m$, there is a sequence 
$n_{i_{1}},n_{i_{2}},\dots,n_{i_{k}}$, where 
$1\leq i_{1} < i_{2} < \cdots < i_{k}\leq 1995$, 
$n_{i_{1}} + \cdots + n_{i_{k}} = q$ and $k$ depends on $q$.

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