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\noindent {\bf 7th IMO 1965}
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\noindent {\bf A1}. Find all $x$ in the interval $[0,2\pi]$ which satisfy: $$2\cos x\le|\sqrt{1+\sin{2x}}-\sqrt{1-\sin{2x}}|\le\sqrt2.$$
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\noindent {\bf A2}. The coefficients $a_{ij}$ of the following equations

$a_{11}x_1+a_{12}x_2+a_{13}x_3=0$

$a_{21}x_1+a_{22}x_2+a_{23}x_3=0$

$a_{31}x_1+a_{32}x_2+a_{33}x_3=0$

\noindent satisfy the following: (a) $a_{11},a_{22},a_{33}$ are positive, (b) other $a_{ij}$ are negative, (c) the sum of the coefficients in each equation is positive. Prove that the only solution is $x_1=x_2=x_3=0$.
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\noindent {\bf A3}. The tetrahedron $ABCD$ is divided into two parts by a plane parallel to $AB$ and $CD$. The distance of the plane from $AB$ is $k$ times its distance from $CD$. Find the ratio of the volumes of the two parts.
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\noindent {\bf B1}. Find all sets of four real numbers such that the sum of any one and the product of the other three is $2$.
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\noindent {\bf B2}. The triangle $OAB$ has $\angle O$ acute. $M$ is an arbitrary point on $AB$. $P$ and $Q$ are the feet of the perpendiculars from $M$ to $OA$ and $OB$ respectively. What is the locus of $H$, the orthocenter of the triangle $OPQ$ (the point where its altitudes meet)? What is the locus if $M$ is allowed to vary over the interior of $OAB$?
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\noindent {\bf B3}. Given $n>2$ points in the plane, prove that at most $n$ pairs of points are the maximum distance apart (of any two points in the set).
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\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

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