Baltic Way 2017 · Problem 9

At first we note that $n=3 k$ is danish for any positive integer $k$, because a hexagon can be cut in 3 equal parallelograms each of which can afterwards be cut in $k$ equal parallelograms (Fig 1).

Furthermore a hexagon can be cut into two equal trapezoids (Fig. 2) each of which can afterwards be cut into 4 equal trapezoids of the same shape (Fig. 3) and so on. Therefore any number of the form $n=2 \cdot 4^{k}$ is also danish.

Figure 1

Figure 2

Figure 3

If we take any danish number $n=2 \cdot 4^{k}$ of the second type, then

$$2^{n}+n=2^{2 \cdot 4^{k}}+2 \cdot 4^{k} \equiv 1+2 \equiv 0(\bmod 3)$$

showing that $2^{n}+n$ is also a danish number.