азицйна T.v.

Petya had several hundred-ruble bills, and no other money. Petya started buying books (each book costs an integer number of rubles) and receiving change in small denominations (1-ruble coins). When buying an expensive book (not cheaper than 100 rubles), Petya paid only with hundred-ruble bills (the minimum necessary number of them), and when buying a cheap book (cheaper than 100 rubles), he paid with small denominations if he had enough, and if not, with a hundred-ruble bill. By the time the hundred-ruble bills ran out, Petya had spent exactly half of his money on books. Could Petya have spent at least 5000 rubles on books?
Let a certain item be bought for $x$ rubles. Increase the cost of these purchases by corresponding amounts, totaling $x$ rubles, and cancel this purchase. Similarly, get rid of all purchases for small change. At each step, the amount of small change decreases, so no new purchases for small change will appear.

There is a purchase worth no more than 50 rubles (small), otherwise, there would be less than half of all the money left. There is only one small purchase, as a second small purchase would have been made from the change of the first, and there are no more purchases for small change now.

The difference between the change for a small purchase and its price is no more than $99-1=98$ rubles. For each other purchase, this difference is negative and even (since the sum of the price and the change is a multiple of 100, i.e., even). Therefore, this difference is no less than -98 and no more than -2. Thus, there are no more than $98: 2=49$ other purchases, and for each of them, no more than two hundred-ruble bills were paid (otherwise, the specified difference is no more than 99 - 201 <-98). Therefore, the total number of hundred-ruble bills was no more than $1+2+\ldots+49=99$, and half of this sum is no more than 9900:2 = $4950<$ 5000.

## Answer

Could not.

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