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\noindent {\bf 3rd IMO 1961}
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\noindent {\bf A1}. Solve the following equations for $x,y$ and $z$: $$x+y+z=a; x^2+y^2+z^2=b^2; xy=z^2.$$ What conditions must $a$ and $b$ satisfy for $x,y$ and $z$ to be distinct positive numbers?
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\noindent {\bf A2}. Let $a,b,c$ be the sides of a triangle and $A$ its area. Prove that: $$a^2+b^2+c^2\ge4\sqrt3A$$ When do we have equality?
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\noindent {\bf A3}. Solve the equation $\cos^nx-\sin^nx=1$, where $n$ is a natural number.
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\noindent {\bf B1}. $P$ is inside the triangle $ABC$. $PA$ intersects $BC$ in $D,PB$ intersects $AC$ in $E$, and $PC$ intersects $AB$ in $F$. Prove that at least one of ${AP\over PD}, {BP\over PE},{CP\over PF}$ does not exceed $2$, and at least one is not less than $2$.
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\noindent {\bf B2}. Construct the triangle $ABC$, given the lengths $AC=b,AB=c$ and the acute $\angle AMB=\alpha$, where $M$ is the midpoint of $BC$. Prove that the construction is possible iff $$b\tan{\alpha\over2}\le c<b.$$ When does equality hold?
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\noindent {\bf B3}. Given three non-collinear points $A,B,C$ and a plane $p$ not parallel to $ABC$ and such that $A,B,C$ are all on the same side of $p$. Take three arbitrary points $A',B',C'$ in $p$. Let $A'',B'',C''$ be the midpoints of $AA',BB',CC'$ respectively, and let $O$ be the centroid of $A'',B'',C''$. What is the locus of $O$ as $A',B',C'$ vary?
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\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

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