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\noindent {\bf 33rd IMO 1992}
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\noindent {\bf A1}. Find all integers $a,b,c$ satisfying $1<a<b<c$ such that $(a-1)(b-1)(c-1)$ is a divisor of $abc-1$.
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\noindent {\bf A2}. Find all functions $f$ defined on the set of all real numbers with real values, such that $f(x^2+f(y))=y+f(x)^2$ for all $x,y$.
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\noindent {\bf A3}. Consider $9$ points in space, no $4$ coplanar. Each pair of points is joined by a line segment which is colored either blue or red or left uncolored. Find the smallest value of $n$ such that whenever exactly $n$ edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color.
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\noindent {\bf B1}. $L$ is a tangent to the circle $C$ and $M$ is a point on $L$. Find the locus of all points $P$ such that there exist points $Q$ and $R$ on $L$ equidistant from $M$ with $C$ the incircle of the triangle $PQR$.
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\noindent {\bf B2}. Let $S$ be a finite set of points in three-dimensional space. Let $S_x,S_y,S_z$ be the sets consisting of the orthogonal projections of the points of $S$ onto the $yz$-plane, $zx$-plane, $xy$-plane respectively. Prove that: $$|S|^2\le |S_x||S_y||S_z|,$$ where $|A|$ denotes the number of points in the set $A$. The orthogonal projection of a point onto a plane is the foot of the perpendicular from the point to the plane.
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\noindent {\bf B3}. For each positive integer $n,S(n)$ is defined as the greatest integer such that for every positive integer $k\le S(n),n^2$ can be written as the sum of $k$ positive squares.

(a) Prove that $S(n)\le n^2-14$ for each $n\ge4$.

(b) Find an integer $n$ such that $S(n)=n^2-14$.

(c) Prove that there are infinitely many integers $n$ such that $S(n)=n^2-14$.
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\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

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