IMO 1981

Problem 1. $P$ is a point inside a given triangle $ABC.D,E,F$ are the feet of the perpendiculars from $P$ to the lines $BC,CA,AB$ respectively. Find all $P$ for which

$\frac{BC}{PD}+\frac{CA}{PE}+\frac{AB}{PF}$

is least.

Problem 2. Let $1\le r\le n$ and consider all subsets of $r$ elements of the set $\{1,2,...,n\}$. Each of these subsets has a smallest member. Let $F(n,r)$ denote the arithmetic mean of these smallest numbers; prove that

$F(n,r)=\frac{n+1}{r+1}.$

Problem 3. Determine the maximum value of $m^3+n^3$,where $m$ and $n$ are integers satisfying $m,n\in \{1,2,...,1981\}$ and $(n^2-mn-m^2{)}^2=1$.

Problem 4. (a) For which values of $n>2$ 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 values of $n>2$ is there exactly one set having the stated property?

Problem 5. Three congruent circles 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 and the point $O$ are collinear.

Problem 6. The function $f(x,y)$ satisfies

(1) $f(0,y)=y+1$,

(2)$f(x+1,0)=f(x,1)$,

(3) $f(x+1,y+1)=f(x,f(x+1,y))$,

for all non-negative integers $x,y$. Determine $f(4,1981)$.