2. a) Determine the integers $a, b, c$ that satisfy the relation: $a^{2}-a+|b-3|+|c-9|=0$.

b) Determine the rational numbers $x, y, z$ that satisfy the relations: $|x-y|=2|y-z|=3|z-x|$ and $x+y+z=2014$.
2. a) If $a \in \mathbb{Z}$, then $a^{2}-a=a(a-1) \geq 0$

It follows that $a^{2}-a=0, b-3=0, c-9=0, a \in\{0,1\}, b=3, c=9$

b) Let $|x-y|=2|y-z|=3|z-x|=k$, then $x-y= \pm k, y-z= \pm \frac{k}{2}, z-x= \pm \frac{k}{3}$

Adding these relations will result in: $0=k\left( \pm 1 \pm \frac{1}{2} \pm \frac{1}{3}\right)$. The expression in parentheses is non-zero, so $k=0$

It follows that $x=y=z=\frac{2014}{3}$