Bogdanov I.I.

On the reverse sides of 2005 cards, different numbers are written (one on each). In one question, it is allowed to point to any three cards and find out the set of numbers written on them. What is the minimum number of questions needed to find out what numbers are written on each card?

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Let there be $N$ questions asked. It is clear that each card participates in at least one question; otherwise, we would not be able to determine the number on it.

Suppose there are $k$ cards that participate in exactly one question.

Then, two such cards cannot appear in the same question. Indeed, if two such cards were to appear in the same question, swapping the numbers on these cards would not change the answers to the questions; therefore, it would be impossible to determine which number is on which card.

Therefore, $k \leq N$. The remaining cards participated in at least two questions. Now, summing the number of questions each card participated in, we get three times the number of questions. Therefore, $3N \geq k + 2(2005 - k) = 4010 - k \geq 4010 - N$, from which $2N \geq 2005$, and $N \geq 1003$.

We will now show a way to determine the numbers with 1003 questions. Set aside one card, and divide the rest into 334 groups of 6 cards each. In each group, number the cards from 1 to 6 and ask three questions: $(1,2,3)$, $(3,4,5)$, and $(5,6,1)$.

Then, the numbers on cards 1, 3, and 5 appear in two answers (for different cards, in different pairs) and are therefore uniquely determined, while the numbers on cards 2, 4, and 6 are the remaining numbers in each of the answers. Thus, in $\frac{2004}{6} \times 3 = 1002$ questions, we can determine the numbers on 2004 cards.

It remains to ask about the set-aside card together with any two already known cards.

## Answer

In 1003 questions.