OBM 1995

$ABCD$ is a quadrilateral with a circumcircle centre $O$ and an inscribed circle centre $I$ . The diagonals intersect at $S$ . Show that if two of $O,I,S$ coincide, then it must be a square.

Find all real-valued functions on the positive integers such that $f(x + 1019) = f(x)$ for all $x$ , and $f(xy) = f(x) f(y)$ for all $x,y$ .

For any positive integer $n>1$ , let $P\left(n\right)$ denote the largest prime divisor of $n$ . Prove that there exist infinitely many positive integers $n$ for which $$P\left(n\right)<P\left(n+1\right)<P\left(n+2\right).$$

A regular tetrahedron has side $L$ . What is the smallest $x$ such that the tetrahedron can be passed through a loop of twine of length $x$ ?

Show that no one $n$ -th root of a rational (for $n$ a positive integer) can be a root of the polynomial $x^5 - x^4 - 4x^3 + 4x^2 + 2$ .

$X$ has $n$ elements. $F$ is a family of subsets of $X$ each with three elements, such that any two of the subsets have at most one element in common. Show that there is a subset of $X$ with at least $\sqrt{2n}$ members which does not contain any members of $F$ .