Question bank IMO 1981

Find the point $P$ inside the triangle $ABC$ for which
$$\frac{BC}{PD}+\frac{CA}{PE}+\frac{AB}{PF}$$
is minimal, where $PD,PE,PF$ are the perpendiculars from $P$ to $BC,CA,AB$ respectively.

Let $f(n,r)$ be the arithmetic mean of the minima of all $r$-subsets of the set $\{1,2,\ldots,n\}$. Prove that $f(n,r)=\frac{n+1}{r+1}$.

Determine the maximum value of $m^2+n^2$ where $m$ and $n$ are integers satisfying $m,n\in\{1,2,\ldots,1981\}$ and $(n^2-mn-m^2)^2=1$.

(a) For which values of $n>2$ is there a set of $n$ consecutive positive integers such that the largest number in the set in the set is a divisor of the least common multiple of the remaining $n-1$ numbers?

(b) For which values of $n>2$ is there a unique set having the stated property?

Three equal circles touch the sides of a triangle and have one common point $O$. Show that the center of the circle inscribed in and of the circle circumscribed about the triangle $ABC$ and the point $O$ are collinear.

Assume that $f(x,y)$ is defined for all positive integers $x$ and $y$, and that the following equations are satisfied:
$$f(0,y)=y+1,\qquad f(x+1,0)=f(x,1),\qquad f(x+1,y+1)=f(x,f(x+1,y)).$$
Determine $f(4,1981)$.