Question bank VMO 1978

Find all three-digit numbers $\overline{xyz}$ such that twice this number equals the sum of $\overline{yzx}$ and $\overline{zxy}$.

Find all values of $m$ for which the following system has exactly one solution:
$$\begin{cases}x^2=2^{|x|}+|x|-y-m,\\x^2=1-y^2.\end{cases}$$

Three food shops $A,B,C$ form a triangle $ABC$ with $\angle CAB=30^\circ$ and $AB=0.75AC$. They regularly supply noodles to the local population in the ratio $5:4:3$, respectively. Where should a noodle-weighing station $M$ be placed to minimize transport costs, assuming the cost is proportional to both the quantity and the distance transported?

Find three fractions in lowest terms $a/d,b/d,c/d$ which form an arithmetic progression and satisfy
$$\frac ba=\frac{1+a}{1+d},\qquad\frac cb=\frac{1+b}{1+d}.$$

A canal has an initial section of width $a$ metres and makes a right-angle turn into a section of width $b$ metres. Find the greatest length of a rectangular raft of width $c$ metres that can float from the first section into the second without getting stuck at the bend.

Let $ABCD.A'B'C'D'$ be a rectangular box. Prove that a triangle can be constructed whose sides equal the distances from $A,A',D$ to the space diagonal $BD'$. Denote the box dimensions by $a_1,a_2,a_3$ and these distances by $m_1,m_2,m_3$. Determine a relation between $a_1,a_2,a_3,m_1,m_2,m_3$.