Question 1
In a convex quadrangle with area $32\,\mathrm{cm}^2$, the sum of the lengths of two nonadjacent edges and of the length of one diagonal is equal to $16\,\mathrm{cm}$. What is the length of the other diagonal?
IMO 1976 — Official contest
Question 2
Let $P_1(x)=x^2-2$, $P_j(x)=P_1(P_{j-1}(x))$, $j=2,3,\ldots$. Show that for arbitrary $n$, the roots of the equation $P_n(x)=x$ are real and different from one another.
Question 3
A rectangular box can be filled completely with unit cubes. If one places cubes with volume $2$ in the box such that their edges are parallel to the edges of the box, one can fill exactly $40\%$ of the box. Determine all possible (interior) sizes of the box.
Question 4
Find the largest number obtainable as the product of positive integers whose sum is $1976$.
Question 5
Let a set of $p$ equations be given,
$$\begin{aligned}a_{11}x_1+\cdots+a_{1q}x_q&=0,\\a_{21}x_1+\cdots+a_{2q}x_q&=0,\\&\vdots\\a_{p1}x_1+\cdots+a_{pq}x_q&=0,\end{aligned}$$
with coefficients $a_{ij}$ satisfying $a_{ij}=-1,0$, or $+1$ for all $i=1,\ldots,p$ and $j=1,\ldots,q$. Prove that if $q=2p$, there exists a solution $x_1,\ldots,x_q$ of this system such that all $x_j$ ($j=1,\ldots,q$) are integers satisfying $|x_j|\le q$ and $x_j\ne0$ for at least one value of $j$.
$$\begin{aligned}a_{11}x_1+\cdots+a_{1q}x_q&=0,\\a_{21}x_1+\cdots+a_{2q}x_q&=0,\\&\vdots\\a_{p1}x_1+\cdots+a_{pq}x_q&=0,\end{aligned}$$
with coefficients $a_{ij}$ satisfying $a_{ij}=-1,0$, or $+1$ for all $i=1,\ldots,p$ and $j=1,\ldots,q$. Prove that if $q=2p$, there exists a solution $x_1,\ldots,x_q$ of this system such that all $x_j$ ($j=1,\ldots,q$) are integers satisfying $|x_j|\le q$ and $x_j\ne0$ for at least one value of $j$.
Question 6
For all positive integral $n$, $u_{n+1}=u_n(u_{n-1}^2-2)-u_1$, $u_0=2$, and $u_1=2\frac12$. Prove that
$$3\log_2[u_n]=2^n-(-1)^n,$$
where $[x]$ is the integral part of $x$.
$$3\log_2[u_n]=2^n-(-1)^n,$$
where $[x]$ is the integral part of $x$.