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\noindent {\bf 1st IMO 1959}
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\noindent {\bf A1}. Prove that $21n+4\over14n+3$ is irreducible for every natural number $n$.
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\noindent {\bf A2}. For what real values of $x$ is $\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=A$ given $A=\sqrt2$, (b) $A=1$, (c) $A=2$, where only non-negative real numbers are allowed in square roots and the root always denotes the non-negative root?
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\noindent {\bf A3}. Let $a,b,c$ be real numbers. Given the equation for $\cos x$: $$a\cos^2x+b\cos x+c=0,$$ form a quadratic equation in $\cos{2x}$ whose roots are the same values of $x$. Compare the equations in $\cos x$ and $\cos{2x}$ for $a=4,b=2,c=-1$.
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\noindent {\bf B1}. Given the length $|AC|$, construct a triangle $ABC$ with $\angle ABC=90^o$, and the median $BM$ satisfying $BM^2=AB\cdot BC$.
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\noindent {\bf B2}. An arbitrary point $M$ is taken in the interior of the segment $AB$. Squares $AMCD$ and $MBEF$ are constructed on the same side of $AB$. The circles circumscribed about these squares, with centers $P$ and $Q$, intersect at $M$ and $N$. 

(a) prove that $AF$ and $BC$ intersect at $N$;

(b) prove that the lines $MN$ pass through a fixed point $S$ (independent of $M$);

(c) find the locus of the midpoints of the segments $PQ$ as $M$ varies.
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\noindent {\bf B3}. The planes $P$ and $Q$ are not parallel. The point $A$ lies in $P$ but not $Q$, and the point $C$ lies in $Q$ but not $P$. Construct points $B$ in $P$ and $D$ in $Q$ such that the quadrilateral $ABCD$ satisfies the following conditions: (1) it lies in a plane, (2) the vertices are in the order $A,B,C,D$, (3) it is an isosceles trapezoid with $AB$ parallel to $CD$ (meaning that $AD=BC$, but $AD$ is not parallel to $BC$ unless it is a square), and (4) a circle can be inscribed in $ABCD$ touching the sides.
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\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

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