1. (10 points) The creative competition at the institute consisted of four tasks. In total, there were 70 applicants. The first test was successfully passed by 35, the second by 48, the third by 64, and the fourth by 63 people, with no one failing all 4 tasks. Those who passed both the third and fourth tests were admitted to the institute. How many were admitted?
Solution. 1st and 2nd tasks were solved by at least $35+48-70=13$ people. 3rd and 4th - at least $64+63-70=57$ people. No one failed all tasks, so 1st and 2nd were solved by 13 people, 3rd and 4th - 57 people.

Answer: 57 people.

Criteria.

| Points | Conditions for awarding |
| :--- | :--- |
| 10 points | Justified solution |
| 5 points | An arithmetic error or insufficiently justified solution in an otherwise justified solution. |
| 0 points | Any other situation |