Boogonov I.I.

The Tsar summoned two sages. He gave the first 100 blank cards and ordered him to write a positive number on each (the numbers do not have to be different), without showing them to the second. Then the first can tell the second several different numbers, each of which is either written on some card or equal to the sum of numbers on some cards (without specifying how each number was obtained). The second must determine which 100 numbers are written on the cards. If he cannot, both will have their heads cut off; otherwise, a number of hairs will be plucked from each beard equal to the number of numbers the first told the second. How can the sages, without communicating, stay alive and lose the minimum number of hairs?
Let's prove that it is sufficient to lose 101 hairs for each sage. Let the first write the numbers $1, 2, 4, \ldots, 2^{99}$ on the cards, and inform the second of these numbers and their sum. When the second hears the number 1, he will understand that there is a card not exceeding 1. Hearing the next number $2^{k}$, he will understand that there is another card not exceeding $2^{k}$, since the sum of the already evaluated cards does not exceed $1+2+\ldots+2^{k-1}<2^{k}$. When he hears $2^{100}-1$, he will already know that the sum of 100 cards does not exceed this number. Therefore, this is the sum, all estimates turn into equalities, and the second will determine all the cards.

Now let's prove that 100 hairs are not enough. Suppose the second is informed of 100 different numbers, and $a<b$ are two of them. Then all the heard numbers can be written on the cards. But in addition to this set of cards, a set with the number $b-a$ instead of $b$ will also work.