Question 16
Prove that every integer from $1$ to $2019$ can be represented as an arithmetic expression consisting of up to $17$ symbols $2$ and an arbitrary number of additions, subtractions, multiplications, divisions and brackets. The $2$'s may not be used for any other operation, for example to form multi-digit numbers (such as $222$) or powers (such as $2^{2}$).
Valid examples:
$$
\left((2 \times 2+2) \times 2-\frac{2}{2}\right) \times 2=22, \quad(2 \times 2 \times 2-2) \times\left(2 \times 2+\frac{2+2+2}{2}\right)=42
$$
Valid examples:
$$
\left((2 \times 2+2) \times 2-\frac{2}{2}\right) \times 2=22, \quad(2 \times 2 \times 2-2) \times\left(2 \times 2+\frac{2+2+2}{2}\right)=42
$$