Gladkova E.B.

Computers $1,2,3, \ldots, 100$ are connected in a ring (the first with the second, the second with the third, ..., the hundredth with the first). Hackers have prepared 100 viruses, numbered them, and at different times, in a random order, they launch each virus on the computer with the same number. If a virus lands on an uninfected computer, it infects it and moves to the next computer in the chain with a higher number until it reaches an already infected computer (from computer 100, the virus moves to computer 1). Then the virus dies, and this computer is restored. No two viruses land on the same computer at the same time. How many computers will be infected as a result of the attack by these 100 viruses?
Let's prove several statements.

1) Every computer is infected by viruses at least twice.

Consider, for example, computer 1. If the first virus to infect it came from computer 100, then it will also be infected by virus 1. If the first virus to infect computer 1 is virus 1, then the virus that first infected computer 100 (which could be the same virus 1, having circled the entire network) will definitely move to computer 1.

2) All viruses will die.

Indeed, each computer is infected by at least two viruses, and the second virus dies by the condition. Therefore, at least one virus will die on each computer. Since there are as many viruses as there are computers, all viruses will die.

3) No computer can be infected by three viruses.

Consider the computer on which the last of all 100 viruses died (let's say it's computer 1; if several died simultaneously, consider any one of them). Then this virus arrived on it as the second one (otherwise, more than one virus would have died on it), and no more viruses arrived on it afterward. Thus, exactly two viruses arrived on computer 1. But then exactly two viruses also arrived on computer 2: one "native" and one from computer 1. Then exactly two viruses also arrived on computer 3. And so on.

Thus, each computer is infected exactly twice, and after that, the computer recovers. Therefore, after all viruses die, all computers will work normally!

## Answer

There will be no infected computers.