1. In the game "Sportlotto-Shish," the main prize is drawn according to the following rules. Each person in the studio independently writes down any number of different pairs of different integers from the set from 1 to 5. If some participants have written down the same pairs, these participants share the main prize. How many participants should be in the studio to ensure that the prize is definitely awarded?
Solution. A total of different pairs can be formed $5 * 4 / 2=10$. It is taken into account that the digits in the pair are different and the order of the digits within the pair does not matter. Each pair can be written down or not written down by the participant. A total of different sets can be formed $2^{10}=1024$. Among them, there is an empty set (no pair is written down). In total, 1023 sets. For someone to have matching sets, the presence of 1024 people is necessary.

Comment. It is essential to exclude the empty set and then add one person (by the pigeonhole principle). A correct answer without excluding the empty set is rated 1 point.