10. 120 schools each send 20 people to form 20 small teams, with each school having exactly 1 person in each team. Find the smallest positive integer $k$, such that after each team selects $k$ people, among all the selected people, there must be 20 people from the same school.
10. Suppose each small team selects $k$ people, and among the selected, no 20 people come from the same school. Therefore, at most 19 people are selected from each school, so $20 k \leqslant 19 \times 120$, which means $k \leqslant 114$. Thus, if $k \geqslant 115$, then after each small team selects $k$ people, there must be 20 people from the same school among the selected. On the other hand, number the schools sequentially as $1,2, \cdots, 120$. If the $i$-th small team does not select the 6 students from schools numbered $6(i-1)+1, 6(i-1)+2, \cdots, 6(i-1)+6$, but selects the other 114 students $(i=1,2, \cdots, 120)$, then each team selects 114 people and at least one person from each school is not selected, thus there are no 20 people from the same school. Therefore, the smallest positive integer $k=115$.