\[Q(x)=\prod_{i}(x-R(x_{i})),\] were $x_{i}$ are roots of P(x).

Your solution does not work for b when $R(x)$ is not monic. Try a new thing for the part b.

What mean "monic"?

http://mathworld.wolfram.com/MonicPolynomial.html

$Q(x)$ defined monic. Why it don't work?

You see any relation with your Old conjucture Omid Hatami ,Some of the Summer camp participants knew about this and you guys put it in the exam? http://www.mathlinks.ro/Forum/viewtopic.php?t=18474

$Q(x)$ that you defined is monic but its coefficients are not integers when $R(x)$ is not monic.

Your conditions are incorrect, I edit them: a) $P(x),R(x)$ are polynomials with rational coefficients and $P(x)$ is not the zero polynomial. Prove that there exist a non-zero polynomial $Q(x)\in\mathbb Q[x]$ that \[P(x)|Q(R(x))\] b) $P,R$ are polynomial with integer coefficients and $P$ is monic. Prove that there exist a monic polynomial $Q(x)\in\mathbb Z[x]$ that \[P(x)|Q(R(x))\] It is easy to check, that $R(x_{i})$ are algebraic integer numbers (for it don't needed, that R(x) - monic, sufficiently, that R(x) had integer coefficients). Therefore $Q(x)=\prod_{i}(x-R(x_{i}))$ monic and had integer coefficients.

To Rust : Yes, You are right. Your solution is correct. I was out of my mind. To lomos_lupin : I don't see any relations between these two problems.