<s>[INST] A question on greatest common divisor I had this question in the Maths Olympiad today. I couldn't solve the part of the greatest common divisor. Please help me understand how to solve it. The question was this:
Let $P(x)=x^3+ax^2+b$ and $Q(x)=x^3+bx+a$, where $a$ and $b$ are non-zero real numbers. If the roots of $P(x)=0$ are the reciprocals of the roots of $Q(x)=0$, then prove that $a$ and $b$ are integers. Also find the greatest common divisor of $P(2013!+1)$ and $Q(2013!+1)$.
Let the roots of $P(x)=0$ be $\alpha,\; \beta, \;and\;\gamma$. Then we have the following four relations. $$\alpha+\beta+\gamma=-a$$ $$\alpha\beta\gamma=-b$$$$\frac{1}{\alpha\beta}+\frac{1}{\beta\gamma}+\frac{1}{\alpha\gamma}=b$$$$\frac{1}{\alpha\beta\gamma}=-a$$
From these, we get $a=b=1$ So, $P(x)=x^3+x^2+1$ and $Q(x)=x^3+x+1$. Now, how to proceed further?
 [/INST] It was not clear to me how the given four relations of the roots gave that $a=b=1$, so here is a different approach.

We have that
$$
Q(x)=x^3+bx+a=0\tag{1}
$$
and
$$
P\left(\frac1x\right)=\frac1{x^3}+a\frac1{x^2}+b=0\tag{2}
$$
have the same roots. Multiplying $(2)$ by $\dfrac{x^3}{b}$ yields
$$
R(x)=x^3+\frac abx+\frac1b=0\tag{3}
$$
Since $Q$ and $R$ have the same roots, so must their difference,
$$
L(x)=\left(a-\frac ab\right)x+\left(a-\frac1b\right)=0\tag{4}
$$
If $a-\frac ab\ne0$, $L$ has one root, and so must $Q$. If $Q$ has only one root, then because the coefficient of $x^2$ is $0$, that root must be $0$. However, since the roots of $P$ and $Q$ are reciprocals, $0$ can't be a root of either. Contradiction.
Therefore, $a-\frac ab=0$ and $a-\frac1b=0$, which means $a=b=1$.

Thus,
$$
\begin{align}
P(x)&=x^3+x^2+1\\
Q(x)&=x^3+x+1
\end{align}\tag{5}
$$
Furthermore,
$$
\gcd(P(x),Q(x))\mid (2x^2-x+1)P(x)-(2x^2+x-2)Q(x)=3\tag{6}
$$
So, considering $\text{mod }3$:
If $x\equiv0\pmod{3}$, then $P(x)\equiv1\pmod{3}$
If $x\equiv1\pmod{3}$, then $P(x)\equiv Q(x)\equiv0\pmod{3}$
If $x\equiv2\pmod{3}$, then $P(x)\equiv1\pmod{3}$
Therefore,
$$
\gcd(P(x),Q(x))=\left\{\begin{array}{l}
3&\text{if }x\equiv1\pmod{3}\\
1&\text{if }x\not\equiv1\pmod{3}
\end{array}\right.\tag{7}
$$
Since $2013!+1\equiv1\pmod{3}$,
$$
\gcd(P(2013!+1),Q(2013!+1))=3\tag{8}
$$
</s>