49th Mathematical Olympiad in Ukraine

a) The example that it isn't necessary: $a=c=2$, $b=d=4$.

b) It shows that if a positive integer number $n$ has its square $n^2$ as a perfect cube of an integer number, then $n$ itself is a perfect cube of some integer number. Really, consider a factorization of $n$ into prime numbers $n=p_1^{m_1}\dots p_k^{m_k}$, its square is $n^2=p_1^{2m_1}\dots p_k^{2m_k}$, if $p_1^{2m_1}\dots p_k^{2m_k}$ is a perfect cube then for each $i=1,k$ we have to hold the condition $2m_i\equiv 0(\mathrm{mod}3)$ which is equal to the condition $m_i\equiv 0(\mathrm{mod}3)$. Hence, the number $n$ is a perfect cube.

As $\frac{ab\cdot cd\cdot ea}{bc\cdot de}=a^2=a_1^3$, then each of the numbers $a$, $b$, $c$, $d$, $e$ is a perfect cube.