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\noindent {\bf 10th IMO 1968}
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\noindent {\bf A1}. Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.
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\noindent {\bf A2}. Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.
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\noindent {\bf A3}. $a,b,c$ are real with $a$ non-zero. $x_1,x_2,\ldots,x_n$ satisfy the $n$ equations:

$ax_i^2+bx_i+c=x_{i+1}$, for $1\le i<n$

$ax_n^2+bx_n+c=x_1$

\noindent Prove that the system has zero, $1$ or $>1$ real solutions according as $(b-1)^2-4ac$ is $<0,=0$, or $>0$.
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\noindent {\bf B1}. Prove that every tetrahedron has a vertex whose three edges have the right lengths to form a triangle.
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\noindent {\bf B2}. Let $f$ be a real-valued function defined for all real numbers, such that for some $a>0$ we have 

$f(x+a)={1\over2}+\sqrt{f(x)-f(x)^2}$ for all $x$.

\noindent Prove that $f$ is periodic, and give an example of such a non-constant $f$ for $a=1$.
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\noindent {\bf B3}. For every natural number $n$ evaluate the sum $$\Bigl[{n+1\over2}\Bigr]+\Bigl[{n+2\over4}\Bigr]+\Bigl[{n+4\over8}\Bigr]+\ldots+\Bigl[{n+2^k\over2^{k+1}}\Bigr]+\ldots,$$ where $[x]$ denotes the greatest integer $\le x$.
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\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

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