Iranian Mathematical Olympiad

The answer of the problem is yes.
Although the statement is discussed on $13$ students, here is an example for $14$ students, all having a chance of being a team member. (students are labelled by $1, 2, \ldots, 14$.)

Note that the fifth person in each test can become a team member, only if all the other four persons above already have been chosen as a team member. Therefore, to prove that the example works, it suffices to show that the fifth person in each test can become a team member. For each of the students $9, 10, 11, 12, 13, 14$, consider these permutations of tests

$$\begin{array}{l} \text{Student 9: } \overbrace{5 \to 6 \to 2 \to 4 \to 3 \to 1}^{\text{Permutation of Tests}} \implies \overbrace{\{1, 2, 3, 4, 6, 9\}}, \\ \text{Student 10: } 3 \to 4 \to 1 \to 6 \to 5 \to 2 \implies \{1, 2, 3, 5, 7, 10\}, \\ \text{Student 11: } 5 \to 6 \to 2 \to 4 \to 1 \to 3 \implies \{1, 2, 3, 4, 6, 11\}, \\ \text{Student 12: } 1 \to 2 \to 3 \to 5 \to 6 \to 4 \implies \{1, 2, 4, 5, 8, 12\}, \\ \text{Student 13: } 3 \to 4 \to 1 \to 6 \to 2 \to 5 \implies \{1, 2, 3, 5, 7, 13\}, \\ \text{Student 14: } 1 \to 2 \to 3 \to 5 \to 4 \to 6 \implies \{1, 2, 4, 5, 8, 14\}. \end{array}$$