Determine all real sequences $a_1,a_2,\ldots,a_{1995}$ satisfying
$$2\sqrt{a_n-(n-1)}\ge a_{n+1}-(n-1)\qquad(n=1,2,\ldots,1994),$$
and $2\sqrt{a_{1995}-1994}\ge a_1+1$.
Let $a_1,a_2,\ldots,a_n$ be integers between $2$ and $1995$ such that (i) any two terms are relatively prime; (ii) each term is either a prime or a product of primes. Determine the smallest possible value of $n$ guaranteeing that the sequence contains a prime number.
Let $PQRS$ be a cyclic quadrilateral such that the segments $PQ$ and $RS$ are not parallel. Consider the circles through $P,Q$ and the circles through $R,S$. Determine the set $A$ of tangency points of circles in these two sets.
Editorial convention (AI): tangent circles are understood to be distinct and to have exactly one common point. The source also discusses a different convention; both interpretations are explained in the solution.
Let $C$ be a circle of radius $R$ and centre $O$, and $S$ a fixed interior point. 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 $M,N',M',N$ as $A$ moves around the whole circle.
Find the smallest positive integer $k$ for which a function $f:\mathbb Z\to\{1,2,\ldots,k\}$ exists satisfying $f(x)\ne f(y)$ whenever $|x-y|\in\{5,7,12\}$.
