Let $S$ be a fixed circle of radius $R$, with center $O$. A fixed point $P$ lies inside it, with $OP=d\lt R$. A variable quadrilateral $ABCD$ has its vertices on $S$, and its diagonals $AC,BD$ intersect at $P$ and are perpendicular. Determine the largest and smallest possible perimeter of $ABCD$ in terms of $R,d$.
Let $n\gt 1$ be an integer not divisible by $1997$. Put
$$a_m=m+\frac{mn}{1997}\quad(m=1,2,\ldots,1996),$$
$$b_m=m+\frac{1997m}{n}\quad(m=1,2,\ldots,n-1).$$
Arrange all the numbers $a_i,b_j$ in increasing order. Prove that the difference between any two consecutive terms is less than $2$.
How many positive-integer-valued functions $f(n)$ defined on the positive integers satisfy $f(1)=1$ and
$$f(n)f(n+2)=f(n+1)^2+1997$$
for every positive integer $n$?
Let $k=\sqrt[3]{3}$. Find a polynomial $p(x)$ with rational coefficients and of the smallest possible degree such that $p(k+k^2)=3+k$. Does there exist a polynomial $q(x)$ with integer coefficients satisfying $q(k+k^2)=3+k$?
There are $75$ points in a unit cube, no three of which are collinear. Prove that three of them can be chosen to form a triangle of area at most $7/72$.