[b]p1[/b]. Your Halloween took a bad turn, and you are trapped on a small rock above a sea of lava. You are on rock $1$, and rocks $2$ through $12$ are arranged in a straight line in front of you. You want to get to rock $12$. You must jump from rock to rock, and you can either (1) jump from rock $n$ to $n + 1$ or (2) jump from rock $n$ to $n + 2$. Unfortunately, you are weak from eating too much candy, and you cannot do (2) twice in a row. How many different sequences of jumps will take you to your destination?


[b]p2.[/b] Find the number of ordered triples $(p; q; r)$ such that $p, q, r$ are prime, $pq + pr$ is a perfect square and $p + q + r \le 100$.


[b]p3.[/b] Let $x, y, z$ be nonzero complex numbers such that $\frac{1}{x}+\frac{1}{y} + \frac{1}{z} \ne 0$ and
$$x^2(y + z) + y^2(z + x) + z^2(x + y) = 4(xy + yz + zx) = -3xyz.$$ Find $\frac{x^3 + y^3 + z^3}{x^2 + y^2 + z^2}$ .


PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1. Let \( a_n \) be the number of different sequences of jumps that will take you to your destination if you are on rock \( 1 \), you want to get to rock \( n+1 \), and you can only jump in the given ways with the given conditions.
2. We wish to find \( a_{11} \). Taking cases on what the first jump is, we get:
   \[
   a_n = a_{n-1} + a_{n-3} \quad \forall \ n \in \mathbb{Z}_{\geq3}.
   \]
3. Also, it is not difficult to see that:
   \[
   a_1 = 1, \quad a_2 = 2, \quad a_3 = 3.
   \]
4. Hence, we get:
   \[
   a_4 = a_3 + a_1 = 3 + 1 = 4,
   \]
   \[
   a_5 = a_4 + a_2 = 4 + 2 = 6,
   \]
   \[
   a_6 = a_5 + a_3 = 6 + 3 = 9,
   \]
   \[
   a_7 = a_6 + a_4 = 9 + 4 = 13,
   \]
   \[
   a_8 = a_7 + a_5 = 13 + 6 = 19,
   \]
   \[
   a_9 = a_8 + a_6 = 19 + 9 = 28,
   \]
   \[
   a_{10} = a_{9} + a_{7} = 28 + 13 = 41,
   \]
   \[
   a_{11} = a_{10} + a_8 = 41 + 19 = 60.
   \]

The final answer is \(\boxed{60}\).