## THE 1999 ASIAN PACIFIC MATHEMATICAL OLYMPIAD

1. 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.

2. Let $a_1, a_2; ...$ be a sequence of real numbers satisfying $a_{i+j} \leq a_i+a_j$ for all $i, j = 1; 2; ...$ .
Prove that
$a_1+\frac{a_2}{2}+\frac{a_3}{3}+...+\frac{a_n}{n} \geq a_n$
for each positive integer $n$.

3. 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 $BR$.

4. Determine all pairs $(a, b)$ of integers with the property that the numbers $a^2 + 4b$ and $b^2 + 4a$ are both perfect squares.

5. 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 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 $n$.