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\noindent {\bf 23rd IMO 1982}
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\noindent {\bf A1}. The function $f(n)$ is defined on the positive integers and takes non-negative integer values. $f(2)=0,f(3)>0,f(9999)=3333$ and for all $m,n$: $f(m+n)-f(m)-f(n)=0$ or $1$. Determine $f(1982)$.
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\noindent {\bf A2}. A non-isosceles triangle $A_1A_2A_3$ has sides $a_1,a_2,a_3$ with $a_i$ opposite $A_i$. $M_i$ is the midpoint of side $a_i$ and $T_i$ is the point where the incircle touches side $a_i$. Denote by $S_i$ the reflection of $T_i$ in the interior bisector of angle $A_i$. Prove that the lines $M_1S_1,M_2S_2$ and $M_3S_3$ are concurrent.
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\noindent {\bf A3}. Consider infinite sequences $\{x_n\}$ of positive reals such that $x_0=1$ and $x_0\ge x_1\ge x_2\ge\ldots$.

(a) Prove that for every such sequence there is an $n\ge1$ such that: $${x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}\ge3.999.$$

(b) Find such a sequence such that for all $n$: $${x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}<4.$$
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\noindent {\bf B1}. Prove that if $n$ is a positive integer such that the equation $$x^3-3xy^2+y^3=n$$
has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.
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\noindent {\bf B2}. The diagonals $AC$ and $CE$ of the regular hexagon $ABCDEF$ are divided by inner points $M$ and $N$ respectively, so that $${AM\over AC}={CN\over CE}=r.$$ Determine $r$ if $B,M$ and $N$ are collinear.
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\noindent {\bf B3}. Let $S$ be a square with sides length $100$. Let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0A_1,A_1A_2,A_2A_3,\ldots,A_{n-1}A_n$ with $A_0=A_n$. Suppose that for every point $P$ on the boundary of $S$ there is a point of $L$ at a distance from $P$ no greater than $1\over2$. Prove that there are two points $X$ and $Y$ of $L$ such that the distance between $X$ and $Y$ is not greater than $1$ and the length of the part of $L$ which lies between $X$ and $Y$ is not smaller than $198$.
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\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

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