[Problems on percentages and ratios $]$ Dirichlet's Principle (etc.) $\quad]$

## Author: : Sergei I.n

A council of 2000 deputies decided to approve the state budget, which contains 200 expenditure items. Each deputy prepared their own budget proposal, in which they indicated the maximum allowable amount of expenditure for each item, ensuring that the total amount of expenditures did not exceed a given amount S. For each item, the council approves the highest amount of expenditure that at least $k$ deputies agree to allocate. What is the smallest $k$ that can guarantee that the total amount of approved expenditures will not exceed $S$?
If $k \leq 1990$, it may happen that the first 10 deputies propose to allocate nothing for the first item of expenditure and allocate $S / 199$ for the rest. The next 10 deputies will allocate nothing for the second item and allocate $S / 199$ for the rest, and so on. As a result, an expenditure amount of $S / 199$ will be approved for each item, and for all two hundred items, the total will be $200 / 199(>$.

If $k=1991$, then after the approval of expenditures for all items, it will turn out that for each item, no more than nine deputies proposed an expenditure amount less than the approved one. Since $200 \cdot 9 < 2000$, there will be a deputy who proposed an expenditure amount not less than the approved one for all items. But the total amount of expenditures proposed by him is no more than $S$, and therefore, the approved total amount is also no more than $S$.

## Answer

$k=1991$.