APMO 1995

Find all real sequences $x_1$, $x_2$, ... , $x_{1995}$ which satisfy 2√($x_n$ - n + 1) ≥ $x_{n+1}$ - n + 1 for n = 1, 2, ... , 1994, and 2√($x_{1995}$ - 1994) ≥ $x_1$ + 1.

Let $a_1$ , $a_2$ , $\ldots$ , $a_n$ be a sequence of integers with values between 2 and 1995 such that: (i) Any two of the $a_i$ 's are relatively 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.

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.

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.

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 $|x-y| \in \{5, 7, 12\}$ .