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\noindent {\bf 18th IMO 1976}
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\noindent {\bf A1}. A plane convex quadrilateral has area $32$, and the sum of two opposite sides and a diagonal is $16$. Determine all possible lengths for the other diagonal.
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\noindent {\bf A2}. Let $P_1(x)=x^2-2$, and $P_{i+1}=P_1(P_i(x))$ for $i=1,2,3,\ldots$. Show that the roots of $P_n(x)=x$ are real and distinct for all $n$.
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\noindent {\bf A3}. A rectangular box can be completely filled with unit cubes. If one places as many cubes as possible, each with volume 2, in the box, with their edges parallel to the edges of the box, one can fill exactly 40\% of the box. Determine the possible dimensions of the box.
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\noindent {\bf B1}. Determine the largest number which is the product of positive integers with sum $1976$.
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\noindent {\bf B2}. $n$ is a positive integer and $m=2n$. $a_{ij}=0,1$ or $-1$ for $1\le i\le n,1\le j\le m$. The $m$ unknowns $x_1,x_2,\ldots,x_m$ satisfy the $n$ equations: $$a_{i1}x_1+a_{i2}x_2+\ldots+a_{im}x_m=0,$$ for $i=1,2,\ldots,n$. Prove that the system has a solution in integers of absolute value at most $m$, not all zero.
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\noindent {\bf B3}. The sequence $u_0,u_1,u_2,\ldots$ is defined by: $u_0,u_1={5\over2},u_{n+1}=u_n(u_{n-1}^2-2)-u_1$ for $n=1,2,\ldots$. Prove that $[u_n]=2^{2^n-(-1)^n\over3}$, where $[x]$ denotes the greatest integer less than or equal to $x$.
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\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

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