Problem 4. The numbers

$$
x=a-2b+2010, y=2b-3c+2010 \text{ and } z=3c-a+2010
$$

are consecutive integers. Determine $x, y$ and $z$.
Solution. Let

$$
a-2 b+2010=k-1 \quad 2 b-3 c+2010=k \text { and } 3 c-a+2010=k+1
$$

where $k \in \mathbb{Z}$. If we add these identities, we get $3 k=3 \cdot 2010$ from which it follows that $k=2010$. from which it is easy to obtain that

$$
x=2009, y=2010 \text { and } z=2011
$$