\hfill \thepage} %} \input{tcilatex} \begin{document} \section{Eighteenth International Olympiad, 1976} \subsection{1976/1.} In a plane convex quadrilateral of area $32,$ the sum of the lengths of two opposite sides and one diagonal is $16.$ Determine all possible lengths of the other diagonal. \subsection{1976/2. } Let $P_{1}(x)=x^{2}-2$ and $P_{j}(x)=P_{1}(P_{j-1}(x))$ for $j=2,3,\cdots $. Show that, for any positive integer $n,$ the roots of the equation $% P_{n}(x)=x$ are real and distinct. \subsection{1976/3. } A rectangular box can be filled completely with unit cubes. If one places as many cubes as possible, each with volume $2,$ in the box, so that their edges are parallel to the edges of the box, one can fill exactly $40\%$ of the box. Determine the possible dimensions of all such boxes. \subsection{1976/4. } Determine, with proof, the largest number which is the product of positive integers whose sum is $1976.$ \subsection{1976/5. } Consider the system of $p$ equations in $q=2p$ unknowns $x_{1},x_{2},\cdots ,x_{q}:$% \begin{eqnarray*} a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1q}x_{q} &=&0 \\ a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2q}x_{q} &=&0 \\ &&\cdots \\ a_{p1}x_{1}+a_{p2}x_{2}+\cdots +a_{pq}x_{q} &=&0 \end{eqnarray*} with every coefficient $a_{ij}$ member of the set $\{-1,0,1\}.$ Prove that the system has a solution $(x_{1},x_{2},\cdots ,x_{q})$ such that (a) all $x_{j}\;(j=1,2,...,q)$ are integers, (b) there is at least one value of $j$ for which $x_{j}\neq 0,$ (c) $\left| x_{j}\right| \leq q(j=1,2,...,q).$ \subsection{1976/6. } A sequence $\{u_{n}\}$ is defined by \[ u_{0}=2,u_{1}=5/2,u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1}\text{for }n=1,2,\cdots \] Prove that for positive integers $n,$% \[ \left[ u_{n}\right] =2^{\left[ 2^{n}-(-1)^{n}\right] /3} \] where $[x]$ denotes the greatest integer $\leq x.$ \end{document}