5. On the surface of a spherical planet, there are $2 n$ gas stations, whose positions are symmetric about the center of the sphere (i.e., for each gas station, there is a corresponding gas station at its symmetric point about the center). Each gas station stores a certain amount of fuel, and the fuel is non-renewable. It is known that a car with an initially empty (and infinitely large) fuel tank can start from any gas station, choose an appropriate route, refuel at the gas stations along the way, and reach the gas station symmetric to the starting point. Here, it is assumed that the car consumes a constant amount of fuel per unit length traveled.

Under the above conditions, find the value of $n$ such that the car can definitely traverse all the gas stations on the planet in some way (the starting and ending points are not restricted).


5. $n=1,2,3$.

Let $X^{\prime}$ denote the gas station symmetric to $X$ with respect to the center of the sphere.
When $n=1$, the conclusion is obviously true.
For $n=2$, let $A B=A^{\prime} B^{\prime}$ be the minimum spherical distance between two gas stations.

By the given condition, a car starting from $A$ can reach $A^{\prime}$. Assume this path is $A \rightarrow B^{\prime} \rightarrow A^{\prime}$ (the path $A \rightarrow B \rightarrow A^{\prime}$ is similar). The fuel at $B$ is sufficient for the car to travel to the nearest gas station $A$, thus, $B \rightarrow A \rightarrow B^{\prime} \rightarrow A^{\prime}$ is a feasible path.

For $n=3$, let the six gas stations be $A, A^{\prime}, B, B^{\prime}, C, C^{\prime}$. As shown in Figure 3, let $A B=A^{\prime} B^{\prime}$ be the minimum spherical distance between two gas stations, and assume $B$ is the nearest gas station to $C$. Divide all gas stations into two groups:
$$
S=\{A, B, C\}, S^{\prime}=\left\{A^{\prime}, B^{\prime}, C^{\prime}\right\},
$$

A car starting from any gas station can reach the other group.
(1) If a car starting from $A$ cannot reach $S^{\prime}$ via the path $A \rightarrow B$, then it also cannot reach $S^{\prime}$ via the path $A \rightarrow C$, otherwise, the path $A \rightarrow B \rightarrow C \rightarrow S^{\prime}$ would be feasible (since the fuel at $B$ is not less than the consumption of the path $A \rightarrow B$, and $B C \leqslant A C$). Therefore, a car starting from $A$ can only directly reach $S^{\prime}$. The nearest gas station in $S^{\prime}$ to $A$ is $C^{\prime}$, thus, the path $C \rightarrow B \rightarrow A \rightarrow C^{\prime} \rightarrow B^{\prime} \rightarrow A^{\prime}$ is feasible.

Similarly, if a car starting from $A^{\prime}$ cannot reach $S$ via the path $A^{\prime} \rightarrow B^{\prime}$, then the path $C^{\prime} \rightarrow B^{\prime} \rightarrow A^{\prime} \rightarrow C \rightarrow B \rightarrow A$ is feasible.
(2) If the assumption in (1) does not hold, then the car can first travel directly from $A$ to $B$. Since $S^{\prime}$ is still reachable, and $B C \leqslant B C^{\prime}=d\left(B, S^{\prime}\right)$, the car can continue from $B$ to $C$.
The fuel at $C$ is not less than the consumption of the path $B \rightarrow C$, and
$$
d\left(C, S^{\prime}\right)=C A^{\prime} \leqslant B C^{\prime}=d\left(B, S^{\prime}\right),
$$

Thus, after reaching $C$, $S^{\prime}$ is still reachable, and the nearest gas station in $S^{\prime}$ to $C$ is $A^{\prime}$.

Therefore, the path $A \rightarrow B \rightarrow C \rightarrow A^{\prime} \rightarrow B^{\prime} \rightarrow C^{\prime}$ is feasible.
The above proves that $n=1,2,3$ all satisfy the requirements.
Next, construct a counterexample for $n \geqslant 4$.
As shown in Figure 4, assume the half-circumference of a great circle on the planet is 1.

Place the gas stations $A_{2}, A_{3}, \cdots, A_{n}$ on a great circle, and satisfy
$$
A_{2} A_{3}=A_{3} A_{4}=\cdots=A_{n-1} A_{n}=d<\frac{1}{n-1},
$$

and arrange the position of gas station $A_{1}$ such that
$$
A_{1} A_{3}=d \text {, and } A_{1} A_{2}=A_{1} A_{4} \text {. }
$$

Assume the gas stations $A_{1}, A_{1}^{\prime}, \cdots, A_{n-1}, A_{n-1}^{\prime}$ each store the fuel required to travel a distance $d$, and $A_{n} 、 A_{n}^{\prime}$ each store the fuel required to travel a distance $1-(n-2) d$. Then a car starting from any gas station can reach the symmetric gas station. The corresponding feasible paths are
$$
\begin{array}{ll} 
& A_{1} \rightarrow A_{3} \rightarrow A_{4} \rightarrow \cdots \rightarrow A_{n} \rightarrow A_{2}^{\prime} \rightarrow A_{3}^{\prime} \rightarrow A_{1}^{\prime} \\
\text { and } & A_{i} \rightarrow A_{i+1} \rightarrow \cdots \rightarrow A_{n} \rightarrow A_{2}^{\prime} \rightarrow A_{3}^{\prime} \rightarrow \cdots \rightarrow A_{i}^{\prime},
\end{array}
$$

where $2 \leqslant i \leqslant n$.
However, let
$$
S=\left\{A_{1}, A_{2}, \cdots, A_{n}\right\}, S^{\prime}=\left\{A_{1}^{\prime}, A_{2}^{\prime}, \cdots, A_{n}^{\prime}\right\},
$$
then the fuel at each of $A_{1}, A_{2}, \cdots, A_{n-1}$ is only sufficient for the car to travel to the nearest gas station, and the fuel at $A_{n} 、 A_{n}^{\prime}$ is only sufficient for the car to travel to the other group ($A_{n}$ to $A_{2}^{\prime}$, $A_{n}^{\prime}$ to $A_{2}$). Therefore, to visit all gas stations, a car starting from $S$ can only visit all gas stations in $S$ without using the fuel at $A_{n}$, which is impossible.
A car starting from $S^{\prime}$ is similar.
This provides a counterexample for $n \geqslant 4$.
In summary, the values of $n$ that satisfy the requirements are only $1, 2, 3$.