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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 28 December 1996}

\medskip
{\bf No calculator is allowed}

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\item Let $ABC$ be a triangle and let $D,E$ and $F$ be the midpoints of the sides 
$AB, BC$ and $CA$ respectively. Suppose that the angle bisector of 
$\angle{BDC}$ meets $BC$ 
at the point $M$ and the angle bisector of 
$\angle{ADC}$ meets $AC$ 
at the point $N$. 
Let $MN$ and $CD$ intersect at $O$ and let the line 
$EO$ meet $AC$ at $P$ and the line $FO$ meet $BC$ at $Q$. 
Prove that $CD=PQ$. 

\item Let $a_{n}$ be the number of $n$-digit integers formed by $1,2$ 
and $3$ which do not contain any consecutive 1's. Prove that $a_{n}$ 
is equal to $(\frac{1}{2}+\frac{1}{\sqrt{3}})(\sqrt{3}+1)^{n}$ rounded  
off to the nearest integer.

\item Let $f : {\Bbb R}\longrightarrow {\Bbb R}$ be a function from the set 
${\Bbb R}$ of real numbers to itself. Find all such functions $f$ 
satisfying the two properties:

(a)\ $f(x+f(y)) = y+f(x)$ \ for all $x,y\in {\Bbb R}$,

(b)\ the set ${\displaystyle \left\{\frac{f(x)}{x} \ : \ \mbox{$x$ is a nonzero real 
number}\right\}}$ is finite.
\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 18 January 1997}

\medskip
{\bf No calculator is allowed}

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\item Four integers $a_{0},b_{0},c_{0},d_{0}$ are written on a circle in the 
clockwise direction. In the first step, we replace  
$a_{0},b_{0},c_{0},d_{0}$ by $a_{1},b_{1},c_{1},d_{1}$, where 
$a_{1}=a_{0}-b_{0},b_{1}=b_{0}-c_{0},c_{1}=c_{0}-d_{0},d_{1}=d_{0}-a_{0}$. 
In the second step, we replace  
$a_{1},b_{1},c_{1},d_{1}$ by $a_{2},b_{2},c_{2},d_{2}$, where 
$a_{2}=a_{1}-b_{1},b_{2}=b_{1}-c_{1},c_{2}=c_{1}-d_{1},d_{2}=d_{1}-a_{1}$. 
In general, at the $k$th step, we have numbers 
$a_{k},b_{k},c_{k},d_{k}$ on the circle where 
$a_{k}=a_{k-1}-b_{k-1},b_{k}=b_{k-1}-c_{k-1},c_{k}=c_{k-1}-d_{k-1},
d_{k}=d_{k-1}-a_{k-1}$. After 1997 such replacements, we set 
$a=a_{1997},b=b_{1997},c=c_{1997},d=d_{1997}$. 
Is it possible that all the numbers $|bc-ad|,|ac-bd|,|ab-cd|$ are primes ?  
Justify your answer.


\item For any positive integer $n$, evaluate \ 
$$\sum_{i=0}^{\lfloor\frac{n+1}{2}\rfloor}
\mbox{\small $\left(\!\!\begin{array}{c} n-i+1\\ 
i\end{array}\!\!\right) $} \ ,$$ 
where  $\displaystyle{\mbox{\small $\left(\!\!\begin{array}{c} m\\ 
k\end{array}\!\!\right) $}  = \frac{m!}{k!(m-k)!}}$ and 
$\lfloor\frac{n+1}{2}\rfloor$ 
is the greatest integer less than or equal to $\frac{n+1}{2}$.

\item Suppose the numbers $a_{0},a_{1},a_{2},\dots,a_{n}$ satisfy 
the following conditions:
$$a_{0}=\frac{1}{2}, \ \ a_{k+1}=a_{k}+\frac{1}{n}a_{k}^{2} \ \ 
\mbox{for}\ \ k=0,1,\dots,n-1.$$
Prove that \ ${\displaystyle 1-\frac{1}{n} < a_{n} < 1}$.

\end{list}

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