Problem 9.1. In a store, there are 20 items, the costs of which are different natural numbers from 1 to 20 rubles. The store has decided to have a promotion: when buying any 5 items, one of them is given as a gift, and the customer chooses which item to receive for free. Vlad wants to buy all 20 items in this store, paying as little as possible. How many rubles will he need? (Each of the 20 items is sold in 1 copy.)
Answer: 136.

Solution. Vlad can take advantage of the offer no more than 4 times, so he will get no more than 4 items for free. The total cost of these 4 items does not exceed $17+18+$ $19+20$ rubles. Therefore, the rubles Vlad needs are not less than

$$
(1+2+3+\ldots+20)-(17+18+19+20)=1+2+3+\ldots+16=\frac{16 \cdot 17}{2}=136
$$

We will show that 136 rubles will definitely be enough for him. He can make purchases of items with the following costs: $(1,2,3,4,17),(5,6,7,8,18),(9,10,11,12,19),(13,14,15,16,20)$. If Vlad takes the last item for free in each purchase, he will spend exactly 136 rubles.