Question 83
On the table there are $2013$ cards with $1, 2, \ldots, 2013$ written on them; the cards are face down (the numbers on them cannot be seen). It is allowed to select any set of cards, to ask if the arithmetic mean of the numbers on them is an integer, and to receive a truthful answer.
a) Find all numbers that can be determined with certainty by asking such questions.
b) We want to divide the cards into groups such that the contents of each group as a whole is known although the values of the individual cards in it might not be. (For instance, to find a group of three cards containing $1, 2, 3$ without knowing which number is on which card.) What maximum number of such groups can be obtained?
a) Find all numbers that can be determined with certainty by asking such questions.
b) We want to divide the cards into groups such that the contents of each group as a whole is known although the values of the individual cards in it might not be. (For instance, to find a group of three cards containing $1, 2, 3$ without knowing which number is on which card.) What maximum number of such groups can be obtained?