IMO 1981

Consider a variable point $P$ inside a given triangle $ABC$ . Let $D$ , $E$ , $F$ be the feet of the perpendiculars from the point $P$ to the lines $BC$ , $CA$ , $AB$ , respectively. Find all points $P$ which minimize the sum $${BC\over PD}+{CA\over PE}+{AB\over PF}.$$

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

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

b.) For which $n>2$ is there exactly one set having this property?

Three circles of equal radius have a common point $O$ and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle are collinear with the point $O$ .

The function $f(x,y)$ satisfies: $f(0,y)=y+1, f(x+1,0) = f(x,1), f(x+1,y+1)=f(x,f(x+1,y))$ for all non-negative integers $x,y$ . Find $f(4,1981)$ .