## THE 1995 ASIAN PACIFIC MATHEMATICAL OLYMPIAD

Time allowed: 4 hours
NO calculators are to be used.
Each question is worth seven points.

Question 1
Determine all sequences of real numbers $a_1, a_2, . . . , a_{1995}$ which satisfy:
$2\sqrt{a_n-(n-1)} \geq a_{n+1}-(n-1)$; for $n = 1, 2,..., 1994$;
and $2 \sqrt{a_{1995}-1994} \geq a_1+1$

Question 2
Let $a_1, a_2, . . .$ an be a sequence of integers with values between $2$ and $1995$ such that:
(i) Any two of the $a_i$ ’s are realtively prime,
(ii) Each $a_i$ is either a prime or a product of primes.
Determine the smallest possible values of $n$ to make sure that the sequence will contain a
prime number.

Question 3
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.

Question 4
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.
Question 5
Find the minimum positive integer $k$ such that there exists a function $f$ from the set $Z$ of all integers to ${1, 2,...,k}$ with the property that $f(x) \neq f(y)$ whenever $|x-y| \in {5,7,12}$.