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\noindent {\bf 4th IMO 1962}
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\noindent {\bf A1}. Find the smallest natural number with $6$ as the last digit, such that if the final $6$ is moved to the front of the number it is multiplied by $4$.
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\noindent {\bf A2}. Find all real $x$ satisfying: $\sqrt{3-x}-\sqrt{x+1}>{1\over2}$.
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\noindent {\bf A3}. The cube $ABCDA'B'C'D'$ has upper face $ABCD$ and lower face $A'B'C'D'$ with $A$ directly above $A'$ and so on. The point $x$ moves at a constant speed along the perimeter of $ABCD$, and the point $Y$ moves at the same speed along the perimeter of $B'C'CB$. $X$ leaves $A$ towards $B$ at the same moment as $Y$ leaves $B'$ towards $C'$. What is the locus of the midpoint of $XY$?
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\noindent {\bf B1}. Find all real solutions to $\cos^2x+\cos^2{2x}+\cos^2{3x}=1$.
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\noindent {\bf B2}. Given three distinct points $A,B,C$ on a circle $K$, construct a point $D$ on $K$, such that a circle can be inscribed in $ABCD$.
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\noindent {\bf B3}. The radius of the circumcircle of an isosceles triangle is $R$ and the radius of its inscribed circle is $r$. Prove that the distance between the two centers is $\sqrt{R(R-2r)}$.
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\noindent {\bf B4}. Prove that a regular tetrahedron has five distinct spheres each tangent to its six extended edges. Conversely, prove that if a tetrahedron has five such spheres then it is regular.
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\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

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