\nopagenumbers
\noindent {\bf 8th IMO 1966}
\vskip 25pt
\noindent {\bf A1}. Problems $A,B$ and $C$ were posed in a mathematical contest. $25$ competitors solved at least one of the three. Amongst those who did not solve $A$, twice as many solved $B$ as $C$. The number solving only $A$ was one more than the number solving $A$ and at least one other. The number solving just $A$ equalled the number solving just $B$ plus the number solving just $C$. How many solved just $B$?
\vskip 12pt
\noindent {\bf A2}. Prove that if $BC+AC=\tan{C\over2}(BC\tan A+AC\tan B)$, then the triangle $ABC$ is isosceles.
\vskip 12pt
\noindent {\bf A3}. Prove that a point in space has the smallest sum of the distances to the vertices of a regular tetrahedron iff it is the center of the tetrahedron.
\vskip 12pt
\noindent {\bf B1}. Prove that ${1\over\sin{2x}}+{1\over\sin{4x}}+\ldots+{1\over\sin{2^nx}}=\cot x-\cot{2^nx}$ for any natural number $n$ and any real $x$ (with $\sin{2^nx}$ non-zero).
\vskip 12pt
\noindent {\bf B2}. Solve the equations $$|a_i-a_1|x_1+|a_i-a_2|x_2+|a_i-a_3|x_3+|a_i-a_4|x_4=1,i=1,2,3,4,$$ where $a_i$ are distinct reals.
\vskip 12pt
\noindent {\bf B3}. Take any points $K,L,M$ on the sides $BC,CA,AB$ of the triangle $ABC$. Prove that at least one of the triangles $AML,BKM,CLK$ has area $\le{1\over4}$ area $ABC$.
\vskip 20pt
\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

\bye