NMC 2003

Stones are placed on the squares of a chessboard having 10 rows and 14 columns. There is an odd number of stones on each row and each column. The squares are coloured black and white in the usual fashion. Show that the number of stones on black squares is even. Note that there can be more than one stone on a square.

Find all triples of integers $(x, y, z)$ satisfying

$$x^{3}+y^{3}+z^{3}-3 x y z=2003$$

The point $D$ inside the equilateral triangle $\triangle A B C$ satisfies $\angle A D C=150^{\circ}$. Prove that a triangle with side lengths $|A D|,|B D|,|C D|$ is necessarily a right-angled triangle.

Figure 14 .

Let $\mathbb{R}^{*}=\mathbb{R} \backslash\{0\}$ be the set of non-zero real numbers. Find all functions $f: \mathbb{R}^{*} \rightarrow \mathbb{R}^{*}$ satisfying

$$f(x)+f(y)=f(x y f(x+y))$$

for $x, y \in \mathbb{R}^{*}$ and $x+y \neq 0$.