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\noindent {\bf 32nd IMO 1991}
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\noindent {\bf A1}. Given a triangle $ABC$, let $I$ be the incenter. The internal bisectors of angles $A,B,C$ meet the opposite sides in $A',B',C'$ respectively. Prove that $${1\over4}<{AI\cdot BI\cdot CI\over AA'\cdot BB'\cdot CC'}\le {8\over27}.$$
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\noindent {\bf A2}. Let $n>6$ be an integer and let $a_1,a_2,\ldots ,a_k$ be all the positive integers less than $n$ and relatively prime to $n$. If $$a_2-a_1=a_3-a_2=\ldots a_k-a_{k-1}>0,$$ prove that $n$ must be either a prime or a power of $2$.
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\noindent {\bf A3}. Let $S=\{1,2,3,\ldots ,280\}$. Find the smallest integer $n$ such that each $n$-element subset of $S$ contains five numbers which are pairwise relatively prime.
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\noindent {\bf B1}. Suppose $G$ is a connected graph with $k$ edges. Prove that it is possible to label the edges $1,2,\ldots ,k$ in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is $1$.
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\noindent {\bf B2}. Let $ABC$ be a triangle and $X$ an interior point of $ABC$. Show that at least one of the angles $XAB,XBC,XCA$ is less than or equal to $30^o$.
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\noindent {\bf B3}. Given any real number $a>1$ construct a bounded infinite sequence $x_0,x_1,x_2,\ldots$ such that $|x_i-x_j||i-j|^a\ge 1$ for every pair of distinct $i,j$.
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\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

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