10.3. On the back of 2005 cards, there are 2005 different real numbers written. Each time you ask a question, you can point to any three cards and ask about the set of 3 numbers written on them. How many times do you need to ask at minimum to definitely find out what numbers are written on the back of each card?
10.3.1003 times.

Assume that $N$ questions have been proposed.
Obviously, each card should participate in at least one question; otherwise, it would be impossible to determine the number on that card.

Suppose there are $k$ cards that each participate in exactly one question. Then, in the same question, it is impossible to encounter two such cards. In fact, if two such cards participate in the same question, swapping the numbers written on these two cards would not change the answer received, making it impossible to determine which number is written on each card. Therefore, $k \leqslant N$. The remaining cards each participate in at least two questions. If we sum the number of questions each card participates in, we get:
$$
3 N \geqslant k + 2(2005 - k) = 4010 - k \geqslant 4010 - N.
$$

Thus, $2 N \geqslant 2005$, which means $N \geqslant 1003$.
Next, we will show that the goal can be achieved with 1003 questions.
Remove 1 card and divide the remaining cards into 334 groups, each containing 6 cards. Number the cards in each group from 1 to 6, and ask three questions for each group: $(1,2,3)$, $(3,4,5)$, $(5,6,1)$. This way, in each group, the cards numbered 1, 3, and 5 each appear in two questions, allowing their numbers to be uniquely determined; the numbers on the cards numbered 2, 4, and 6 can also be determined. Thus, through $\frac{2004}{6} \times 3 = 1002$ questions, the numbers on 2004 cards can be determined. The remaining one question is used to determine the number on the card that was set aside (by asking it along with any two other cards).