# Problem 4.

Over two days, 100 bankers collected funds to fight a new virus. Each of them made a one-time contribution of a whole number of thousands of rubles, not exceeding 200. Each contribution on the first day did not exceed 100 thousand, while on the second day it was more than this amount; and no pair of all 100 contributions differed by exactly 100 thousand. What amount was collected?
The solution significantly depends on whether all contributions were distinct or could repeat.

Let's first consider the case where all contributions are distinct. Any natural number greater than 100 but not exceeding 200 can be represented as $100+n$, where $n \in [1,2,3, \ldots, 100]$. According to the condition, there are no two numbers that differ exactly by 100, so $n$ takes all the remaining values from the first 50 numbers. Therefore, the sum of all 100 numbers is $1+2+3+\ldots+100+100 \cdot 50=10050$.

If contributions could repeat, a definitive answer cannot be given, but estimates can be made. Note that according to the condition, there was at least one contribution each day. The smallest sum would occur if there were 99 contributions of 1 on the first day, and one contribution of 102 on the second day (101 is not possible). The total sum would be $99 \cdot 1 + 102 = 201$. The largest sum would occur if there was 1 contribution of 99 on the first day, and 99 contributions of 200 on the second day. The total sum would be $99 \cdot 200 + 99 = 19899$.

Answer. If all contributions are distinct, then the total collected is 10050 thousand. If not, the minimum possible sum is 201 thousand, and the maximum possible sum is 19899 thousand.