IMO 1976

Problem 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.

Problem 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.

Problem 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.

Problem 4. Determine, with proof, the largest number which is the product of positive integers whose sum is $1976$.

Problem 5. Consider the system of $p$ equations in $q=2p$ unknowns $x_1,x_2,\cdots ,x_q:$

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 $x_j\ne 0$,

(c) $\left|x_j\right|\le q(j=1,2,...,q)$.

Problem 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 $\le x$.