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\noindent {\bf 19th IMO 1977}
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\noindent {\bf A1}. Construct equilateral triangles $ABK, BCL, CDM, DAN$ on the inside of a square $ABCD$. Show that the midpoints of $KL,LM,MN,NK$ and the midpoints of $AK,BK,BL,CL,CM,DM,DN,AN$ form a regular dodecahedron.
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\noindent {\bf A2}. In a finite sequence of real numbers the sum of any seven successive terms is negative, and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.
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\noindent {\bf A3}. Given an integer $n>2$, let $V_n$ be the set of integers $1+kn$ for $k$ a positive integer. A number $m$ in $V_n$ is called indecomposable if it cannot be expressed as the product of two members of $V_n$. Prove that there is a number in $V_n$ which can be expressed as the product of indecomposable members of $V_n$ in more than one way (decompositions which differ solely in the order of factors are not regarded as different).
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\noindent {\bf B1}. Define $f(x)=1-a\cos x-b\sin x-A\cos2x-B\sin2x$, where $a,b,A,B$ are real constants. Suppose $f(x)\ge)$ for all real $x$. Prove that $a^2+b^2\le2$ and $A^2+B^2\le1$.
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\noindent {\bf B2}. Let $a$ and $b$ be positive integers. When $a^2+b^2$ is divided by $a+b$, the quotient is $q$ and the remainder is $r$. Find all pairs $a,b$ such that $q^2+r=1977$.
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\noindent {\bf B3}. The function $f$ is defined on the set of positive integers and its values are positive integers. Given that $f(n+1)>f(f(n))$ for all $n$, prove that $f(n)=n$ for all $n$.
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\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

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