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\noindent {\bf 22nd IMO 1981}
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\noindent {\bf A1}. $P$ is a point inside the triangle $ABC$. $D,E,F$ are the feet of the perpendiculars from $P$ to the lines $BC,CA,AB$ respectively. Find all $P$ which minimize: $${BC\over PD}+{CA\over PE}+{AB\over PF}.$$
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\noindent {\bf A2}. Take $r$ such that $1\le r\le n$, and consider all subsets of $r$ elements of the set $\{1,2,\ldots,n\}$. Each subset has a smallest element. Let $F(n,r)$ be the arithmetic mean of these smallest elements. Prove that: $$F(n,r)={n+1\over r+1}.$$
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\noindent {\bf A3}. Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.
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\noindent {\bf B1}. (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?
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\noindent {\bf B2}. 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 $O$.
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\noindent {\bf B3}. 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 $f(4,1981)$.
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\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

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