## Task 4 - 100614

A total of 25 answer cards marked "very well solved" were sent by the editorial team to 15 participants in the mathematics student magazine "alpha" competition, and each of these participants received at least one such answer card.

In addition, it is known about these 15 participants that at least one participant received exactly 2 answer cards, at least one participant received exactly 3 answer cards, at least one participant received exactly 4 answer cards, and at least one participant received exactly 5 answer cards. Some of the 15 participants received exactly one answer card each.

Determine the number of these participants!
According to the task, among the 15 participants, there are at least 4 for whom the following is true:

the first of them received exactly 2 answer cards, the second of them received exactly 3 answer cards, the third of them received exactly 4 answer cards, and the fourth of them received exactly 5 answer cards.

These 4 participants thus received exactly 14 answer cards in total. Since each of the remaining 11 participants received at least one answer card, all 25 answer cards have already been distributed.

None of these 11 participants could therefore have received more than one answer card.

Among the 15 mentioned participants, there are thus exactly 11 with exactly one answer card each.

Solutions of the I. Round 1970 adopted from [5]

### 3.11.2 II. Round 1970, Class 6