IMO Hk TST

a. The smallest possible value of $Q$ is $\frac{1}{3^{2005}}$.
By the power mean inequality, we have
$$S_n \ge 3 \left(\frac{S_1}{3}\right)^n = \frac{1}{3^{n-1}}$$
for any positive integer $n$. Therefore, we have
$$Q = S_3 S_{2004} \ge \frac{1}{3^2} \cdot \frac{1}{3^{2003}} = \frac{1}{3^{2005}}.$$
Equality holds when $x = y = z = \frac{1}{3}$. Therefore, the minimum value of $Q$ is $\frac{1}{3^{2005}}$.

b. $P$ is larger than $Q$.
Indeed, we have
$$\begin{align*} P \ge Q & \Leftrightarrow (x^2 + y^2 + z^2)(x^{2005} + y^{2005} + z^{2005}) \ge (x^3 + y^3 + z^3)(x^{2004} + y^{2004} + z^{2004}) \\ & \Leftrightarrow \sum_{\text{sym}} x^{2005} y^2 \ge \sum_{\text{sym}} x^{2004} y^3. \end{align*}$$

This is true by Muirhead's theorem. Equality holds when $x = y = z$. Since it is given that $x$, $y$, $z$ are pairwise distinct, the inequality is strict, and hence $P > Q$.