Which of the following actions is allowed for district boards on election day?

The action that is allowed for district boards on election day is displaying the flag outside the entrance. This practice is in accordance with the laws and regulations governing election day activities, which emphasize maintaining a neutral and respectful environment within polling places. Displaying the flag not only symbolizes patriotism but also signifies the importance of the democratic process, without infringing on voters' rights or influencing their choices.

Other actions, such as encouraging voters to vote for specific candidates or creating promotional materials, can lead to potential violations of election laws, as they may be seen as attempts to sway or influence the vote. Accepting bribes for votes is illegal and undermines the integrity of the electoral process, thus is strictly prohibited.