Evochkimov M.A.

The company recorded its expenses in rubles under 100 budget items, resulting in a list of 100 numbers (each number has no more than two decimal places). Each accountant took a copy of the list and found an approximate sum of the expenses, acting as follows. First, he arbitrarily chose two numbers from the list, added them, discarded the digits after the decimal point (if any), and recorded the result in place of the two chosen numbers. With the resulting list of 99 numbers, he did the same, and so on, until only one integer remained in the list. It turned out that in the end, all accountants got different results. What is the maximum number of accountants that could have worked in the company?
Evaluation. Accountants each time calculated the integer part of the sum of some two numbers, which is equal to the sum of their integer parts plus, possibly, one. Let's call this addition of one an incident. It could only happen when both addends were non-integers. Each accountant ended up with the sum of the integer parts of the original numbers plus the number of incidents that occurred to him. Since each incident removed two non-integers, there were at most 50 incidents, and 51 different possible outcomes.

Example. Suppose there were 50 numbers 0.3 and 50 numbers 0.7. We will show how an accountant can obtain any integer result $k$ from 0 to 50. First, he adds $0.3+0.7$ $k$ times and gets $k$ ones (for $k=0$, he adds $0.3+0.3$). Then, to the already existing integer, he adds the remaining numbers in sequence without incidents.

## Answer

51 accountants.