This is the 917th most frequent Ukrainian word.
залишаються
"Remain" or "stay."
In this sentence, "залишаються" is used to indicate that the state of the letters is unchanged; they stay unsealed.
Листи залишаються незапечатаними.
The letters remain unsealed.
Here, "залишаються" shows that the students stay in the specific location, i.e., the library, even after it is supposed to be closed.
Студенти залишаються в бібліотеці після закриття.
The students remain in the library after closing.
In this example, "залишаються" is employed to express that old customs continue to hold significance currently.