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{\centerline{\bf THE 1995 ASIAN PACIFIC MATHEMATICAL OLYMPIAD}}
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{\it Time allowed: 4 hours}

{\it NO calculators are to be used.}

{\it Each question is worth seven points.}
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{\bf Question 1}
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Determine all sequences of real numbers $a_1$, $a_2$, \ldots, $a_{1995}$ which
satisfy: $$2\sqrt{a_n - (n - 1)} \geq a_{n+1} - (n - 1), \ \mbox{for} \ n = 1,
2, \ldots 1994,$$ and $$2\sqrt{a_{1995} - 1994} \geq a_1 + 1.$$
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{\bf Question 2}
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Let $a_1$, $a_2$, \ldots, $a_n$ be a sequence of integers with values between 2
and 1995 such that:
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(i) Any two of the $a_i$'s are realtively prime,

(ii) Each $a_i$ is either a prime or a product of primes.
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Determine the smallest possible values of $n$ to make sure that the sequence
will contain a prime number.
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{\bf Question 3}
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Let $PQRS$ be a cyclic quadrilateral such that the segments $PQ$ and $RS$ are
not parallel.  Consider the set of circles through $P$ and $Q$, and the set of
circles through $R$ and $S$.  Determine the set $A$ of points of tangency of
circles in these two sets.
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{\bf Question 4}
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Let $C$ be a circle with radius $R$ and centre $O$, and $S$ a fixed point in
the interior of $C$.  Let $AA'$ and $BB'$ be perpendicular chords through $S$.
Consider the rectangles $SAMB$, $SBN'A'$, $SA'M'B'$, and $SB'NA$.  Find the set
of all points $M$, $N'$, $M'$, and $N$ when $A$ moves around the whole circle.
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{\bf Question 5}
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Find the minimum positive integer $k$ such that there exists a function $f$
from the set $\Bbb{Z}$ of all integers to $\{1, 2, \ldots k\}$ with the
property that $f(x) \neq f(y)$ whenever $|x-y| \in \{5, 7, 12\}$.
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