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\noindent {\bf 25th IMO 1984}
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\noindent {\bf A1}. Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.
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\noindent {\bf A2}. Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$.
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\noindent {\bf A3}. Given points $O$ and $A$ in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point $X$ in the plane, the circle $C(X)$ has center $O$ and radius $OX+{\angle AOX\over OX}$, where $\angle AOX$ is measured in radians in the range $[0,2\pi)$. Prove that we can find a point $X$, not on $OA$, such that its color appears on the circumference of the circle $C(X)$.
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\noindent {\bf B1}. Let $ABCD$ be a convex quadrilateral with the line $CD$ tangent to the circle on diameter $AB$. Prove that the line $AB$ is tangent to the circle on diameter $CD$ iff $BC$ and $AD$ are parallel.
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\noindent {\bf B2}. Let $d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $n>3$ vertices. Let $p$ be its perimeter. Prove that: $$n-3<{2d\over p}<\Bigl[{n\over2}\Bigr]\Bigl[{n+1\over 2}\Bigr]-2,$$ where $[x]$ denotes the greatest integer not exceeding $x$.
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\noindent {\bf B3}. Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.
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\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

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