# Task 5. Maximum 15 points

In the treasury of the Magic Kingdom, they would like to replace all old banknotes with new ones. There are a total of 3,628,800 old banknotes in the treasury. Unfortunately, the machine that prints new banknotes requires major repairs and each day it can produce fewer banknotes: on the first day, it can only produce half of the banknotes that need to be replaced; on the second day, only a third of the remaining old banknotes in the treasury; on the third day, only a quarter, and so on. Each run of the machine in any state costs the treasury 90,000 monetary units (m.u.), and major repairs will cost 800,000 m.u. After major repairs, the machine can produce no more than one million banknotes per day. The kingdom has allocated no more than 1 million m.u. for the renewal of banknotes in the treasury.

(a) After how many days will 80% of the old banknotes be replaced?

(b) Will the kingdom be able to replace all the old banknotes in the treasury?
# Solution:

(a) If a major repair is to be carried out, it is most effective to do so on the second day, as this will allow the production of new banknotes to increase from 604,800 to 1 million on that day, and the production will also be 1 million in subsequent days, which is more than the possibilities without major repairs. It is not efficient to carry out the major repair on the first day, as the production without it is 1,814,400 banknotes. (9 points).

With the major repair, the budget will cover the repair itself and 2 launches of the machine, thus the maximum number of banknotes that can be replaced is 1,814,400 + 1,000,000 = 2,814,400 banknotes, which is less than 80% of the number of old banknotes (2 points).

Let's consider the option of not repairing the machine. We have enough funds for only 11 launches. In percentage terms, the remaining amount is:

After the first day: 50%.

After the second day: 200/3.

After the third day: 25%.

After the fourth day: 20%.

Thus, it will take 4 days (2,903,040 banknotes will be replaced), which is 80% of the banknotes. (2 points)

(b) After 11 launches, we will only be able to replace 3,326,400 banknotes, i.e., in any case, we will not be able to recast all the banknotes without repairing the machine.

It can also be noted that with any launch, there will always be a part of the banknotes that remain unreplaced. (2 points)

# Federal State Budget Educational Institution of Higher Education Russian Presidential Academy of National Economy and Public Administration

Olympiad of Schoolchildren of RANEPA in Economics 2022 - 2023 academic year 10-11 grade Final stage Variant 2.

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