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\noindent {\bf 5th IMO 1963}
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\noindent {\bf A1}. For which real values of $p$ does the equation $$\sqrt{x^2-p}+2\sqrt{x^2-1}=x$$ have real roots? What are the roots?
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\noindent {\bf A2}. Given a point $A$ and a segment $BC$, determine the locus of all points $P$ in space for which $\angle APX=90^o$ for some $X$ on the segment $BC$.
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\noindent {\bf A3}. An $n$-gon has all angles equal and the lengths of consecutive sides satisfy $a_1\ge a_2\ge\ldots\ge a_n$. Prove that all the sides are equal.
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\noindent {\bf B1}. Find all solutions $x_1,\ldots,x_5$ to the five equations $x_i+x_{i+2}=yx_{i+1}$ for $i=1,\ldots,5$, where subscripts are reduced by $5$ if necessary.
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\noindent {\bf B2}. Prove that $\cos{\pi\over7}-\cos{2\pi\over7}+\cos{3\pi\over7}={1\over2}$.
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\noindent {\bf B3}. Five students $A,B,C,D,E$ were placed $1$ to $5$ in a contest with no ties. One prediction was that the result would be the order $A,B,C,D,E$. But no student finished in the position predicted and no two students predicted to finish consecutively did so. For example, the outcome for $C$ and $D$ was not $1,2$ (respectively), or $2,3$, or $3,4$ or $4,5$. Another prediction was the order $D,A,E,C,B$. Exactly two students finished in the places predicted and two disjoint pairs predicted to finish consecutively did so. Determine the outcome.
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\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

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