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\noindent {\bf 6th IMO 1964}
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\noindent {\bf A1}. (a) Find all natural numbers $n$ for which $7$ divides $2^n-1$.

(b) Prove that there is no natural number $n$ for which $7$ divides $2^n+1$.
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\noindent {\bf A2}. Suppose that $a,b,c$ are the sides of a triangle. Prove that: $$a^2(b+c-a)+b^2(c+a-b)+c^2(a+b-c)\le3abc.$$
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\noindent {\bf A3}. Triangle $ABC$ has sides $a,b,c$. Tangents to the inscribed circle are constructed parallel to the sides. Each tangent forms a triangle with the other two sides of the triangle and a circle is inscribed in each of these triangles. Find the total area of all four inscribed circles.
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\noindent {\bf B1}. Each pair from $17$ people exchange letters on one of three topics. Prove that there are at least $3$ people who write to each other on the same topic. [In other words, if we color the edges of the complete graph $K_17$ with three colors, then we can find a triangle all the same color.]
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\noindent {\bf B2}. $5$ points in a plane are situated so that no two of the lines joining a pair of points are coincident, parallel or perpendicular. Through each point lines are drawn perpendicular to each of the lines through two of the other $4$ points. Determine the maximum number of intersections these perpendiculars can have.
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\noindent {\bf B3}. $ABCD$ is a tetrahedron and $D_0$ is the centroid of $ABC$. Lines parallel to $DD_0$ are drawn through $A,B$ and $C$ and meet the planes $BCD,CAD$ and $ABD$ in $A_0,B_0$ and $C_0$ respectively. Prove that the volume of $ABCD$ is one-third of the volume of $A_0B_0C_0D_0$. Is the result true if $D_0$ is an arbitrary point inside $ABC$?
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\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

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