\hfill \thepage} %} \input{tcilatex} \begin{document} \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 $f(4,1981).$ \end{document}