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\begin{center}

${\bf 11^{th}}$ {\bf Asian Pacific Mathematical Olympiad}

\end{center}



\begin{center}

{\bf March, 1999}

\end{center}



\bigskip



\begin{enumerate}



\item

Find the smallest positive integer $n$ with the following property: there

does not exist an arithmetic progression of 1999 real numbers containing

exactly $n$ integers.



\item

Let $a_1, a_2, \dots$ be a sequence of real numbers satisfying $a_{i+j} \leq a_i+a_j$

for all $i,j=1,2,\dots$. Prove that

\[

a_1 + \frac{a_2}{2} + \frac{a_3}{3} + \cdots + \frac{a_n}{n} \geq a_n

\]

for each positive integer $n$.



\item

Let $\Gamma_1$ and $\Gamma_2$ be two circles intersecting at $P$ and $Q$. The common

tangent, closer to $P$, of $\Gamma_1$ and $\Gamma_2$ touches $\Gamma_1$ at $A$ and

$\Gamma_2$ at $B$. The tangent of $\Gamma_1$ at $P$ meets $\Gamma_2$ at $C$,

which is different from $P$, and the extension of $AP$ meets $BC$ at $R$.

Prove that the circumcircle of triangle $PQR$ is tangent to $BP$ and $BR$.



\item

Determine all pairs $(a,b)$ of integers with the property that the numbers $a^2+4b$

and $b^2+4a$ are both perfect squares.



\item

Let $S$ be a set of $2n+1$ points in the plane such that no three are collinear

and no four concyclic. A circle will be called \emph{good} if it has 3 points of $S$

on its circumference, $n-1$ points in its interior and $n-1$ points in its exterior.

Prove that the number of good circles has the same parity as $n$.



\end{enumerate}





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