DE-MO 1996

Find all natural numbers $n$ with the following property: Given the decimal writing of $n$ , adding a few digits one can obtain the decimal writing of $1996n$ .

Let $a$ and $b$ be positive real numbers smaller than $1$ . Prove that the following two statements are equivalent: (i) $a+b = 1$ , (ii) Whenever $x,y$ are positive real numbers such that $x < 1, y < 1, ax+by < 1$ , the following inequlity holds: $$\frac{1}{1-ax-by} \le \frac{a}{1-x} + \frac{b}{1-y}$$

Let be given an arbitrary tetrahedron $ABCD$ with volume $V$ . Consider all lines which pass through the barycenter $S$ of the tetrahedron and intersect the edges $AD,BD,CD$ at points $A',B',C$ respectively. It is known that among the obtained tetrahedra there exists one with the minimal volume. Express this minimal volume in terms of $V$

Find all pairs of real numbers $(x,y)$ which satisfy the system $$\begin{cases} x-y = 7 \sqrt[3]{x^2}+\sqrt[3]{xy}+\sqrt[3]{y^2} = 7\end{cases}$$

Given two non-intersecting chords $AB$ and $CD$ of a circle $k$ and a length $a <CD$ . Determine a point $X$ on $k$ with the following property: If lines $XA$ and $XB$ intersect $CD$ at points $P$ and $Q$ respectively, then $PQ = a$ . Show how to construct all such points $X$ and prove that the obtained points indeed have the desired property.