Ukrajina 2008

a) We can easily find that the GCD of all the numbers $42$, $70$, $105$, $462$, and $2009$ is $7$. For every operation you are allowed to perform the number of stones in piles stays divisible by $7$. Therefore the answer is "No" as the number $2008$ is not divisible by $7$.

b) In the second case the answer is "Yes". One of the ways to obtain the necessary set of piles is as follows: $(42,70,105,462,2009)\to (42,35,105,462,2009)\to (7,35,105,462,2009)$. Now we have a pile of $7$ stones and can increase the other pile by $7$ stones $1061$ times: $\to (7,35+7\times 1061,105,462,2009)$. After that we divide the five derived piles into equal parts, in which the total number of stones equals $10045$, and arrive at the desired result.