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\noindent {\bf 44th IMO 2003}
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\noindent {\bf A1}. $S$ is the set $\{1, 2, 3, \ldots , 1000000\}$. Show that for any subset $A$ of $S$ with 101 elements we can find 100 distinct elements $x_i$ of $S$, such that the sets $\{a+x_i | a \in A\}$ are all pairwise disjoint. 
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\noindent {\bf A2}. Find all pairs $(m, n)$ of positive integers such that ${m^2 \over 2mn^2 - n^3 + 1}$ is a positive integer.
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\noindent {\bf A3}. A convex hexagon has the property that for any pair of opposite sides the distance between their midpoints is $\sqrt{3}/2$ times the sum of their lengths Show that all the hexagon\rq s angles are equal.
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\noindent {\bf B1}. $ABCD$ is cyclic. The feet of the perpendicular from $D$ to the lines $AB, BC, CA$ are $P, Q, R$ respectively. Show that the angle bisectors of $ABC$ and $CDA$ meet on the line $AC$ iff $RP = RQ$.
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\noindent {\bf B2}. Given $n > 2$ and reals $x_1 \le x_2 \le \cdots \le x_n$, show that $(\sum_{i,j} |x_i - x_j|)^2 \le {2 \over 3}(n^2 - 1) \sum_{i,j} (x_i - x_j)^2$. Show that we have equality iff the sequence is an arithmetic progression.
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\noindent {\bf B3}. Show that for each prime $p$, there exists a prime $q$ such that $n^p - p$ is not divisible by $q$ for any positive integer $n$.
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\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

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