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\section{Twenty-second International Olympiad, 1981}

1981/1. $P$ 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.

1981/2. 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 
\[
F(n,r)=\frac{n+1}{r+1}. 
\]

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

1981/4. (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?

1981/5. 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.

1981/6. 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 $f(4,1981).$

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