Ya Chenenko I.V.

In the store, they sell DVD discs - individually and in two types of packages (packages of different types differ in quantity and cost). Vasya calculated how much money is needed to buy $N$ discs (if it is more profitable to buy more discs than needed - Vasya does so):

| $N$ | 1 | 2 | 3 | 4 | 5 | $6-10$ | 11 | 12 | 13 | 14 | 15 | $16-20$ | 21 | 22 | $23-25$ | 26 | 27 | 28 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| c. | 20 | 40 | 60 | 80 | 100 | 111 | 131 | 151 | 171 | 191 | 211 | 222 | 242 | 262 | 265 | 285 | 305 | 325 |

How many discs were in the packages and at what price were the packages sold?

How much money does Vasya need to buy at least 29 discs?
If the most profitable purchase consists of several blocks (disks individually and packages), then the disks contained in any block or any set of these blocks were purchased in a way that cannot be "improved" (made more profitable) - otherwise, by "improving" the payment method for part of the disks, we would be able to make the entire purchase more profitable.

From this, it follows that if we used a certain package for the purchase, then to buy the number of disks contained in this package at the minimum price, we can buy just this package.

From the table, it can be seen that disks in quantities of 1, 2, 3, 4, 5 were purchased individually at a price of 20 rubles per disk. Purchasing no less than 6 disks at this price does not correspond, so in this case, at least one package was purchased. But nothing else, besides this package, could have been purchased, as it would have resulted in this package optimizing the purchase of fewer than 6 disks. But then we would have used such a method to purchase such a (less than 6) quantity.

When purchasing 7 disks, the same package should have been used. The same sequentially (using the previously obtained information) we establish for 8, 9, and 10 disks.

The price of 11 disks is higher than 10. Therefore, this package contains only 10 disks. Thus, we have learned the parameters of one of the packages: it contains 10 disks and costs 111 rubles.

Now notice that the price of purchasing 24 and 25 disks is the same, while for purchasing 26, it is different.

When purchasing 25 disks, we could not use individual disks (since otherwise, by refusing one individual disk, we could have bought 24 disks 20 rubles cheaper than 25 disks, while the table shows the same price for these quantities). And it is impossible to get 25 disks with packages of 10 disks. Therefore, we have established another type of package: it contains 25 disks and costs 265 rubles. This package cannot contain more than 25 disks, as then 26 disks would cost the same.

Thus, we have established 2 types of packages: 1) 111 rubles for 10 disks (11.1 rubles per disk); 2) 265 rubles for 25 disks (10.6 rubles per disk).

Since the condition states that there should be only two types of packages, we have determined all possible options.

Let's determine the optimal price for purchasing 29 disks. We can buy three packages of 10 disks, which will cost 333 rubles.

Purchasing individual disks is irrational: as soon as we buy one individual disk (for 20 rubles), we will need to buy 28 disks, the optimal price of which, according to the condition, is 325 rubles. In total, it will be 345 rubles.

If we buy a package of 25 disks, we will need to buy 4 more disks. From the table, it can be seen that it is cheapest to buy them individually. But, as we just found out, the price will be no less than 345 rubles.

Therefore, the initially considered option (3 packages of 10 disks) is the most

rational.

## Answer

For sale are: individual disks at a price of 20 rubles; packages of 10 disks at a price of 111 rubles per package; packages of 25 disks at a price of 265 rubles per package. To purchase no less than 29 disks, 333 rubles are required.