8.4. In the Go section, there are 55 boys of different ratings. The boys decided to play a tournament, each with each other in one game. To make it more interesting, some of the boys were allowed to use a computer's help exactly once during the tournament. If in a game, one of the boys uses a computer's help and the other does not, the one using the computer wins; otherwise, the one with the higher rating wins. There are no draws in Go.

By the end of the tournament, two boys were found, each of whom won more games than any of the two boys with the highest ratings. What could be the maximum number of boys who did not use a computer?

Answer, option 1. 45 players.

Answer, option 2. 55 players.

Answer, option 3. 35 players.

Answer, option 4. 50 players.
Solution Option 1. Let's number the players in descending order of their strength from first to fiftieth.

We will assume that each won game brings the player one point. Consider two players who ultimately scored more points than the two strongest participants. It is clear that the number of one of them is not less than 4. If he did not use the computer, he could win no more than 46 players. Therefore, even with the help of the computer, he could score no more than 47 points. This means that each of the two strongest players scored no more than 46 points. If no one had used the computer, the two strongest would have scored 49 and 48 points. Thus, due to the computer, they must have lost at least 3 and 2 points, respectively. Therefore, the computer was used no less than \(3+2=5\) times, meaning that no more than 45 boys played without it.

Let's provide an example showing that there could have been 5 boys who used the computer. Suppose players from 3rd to 7th used the computer: three of them won against the 1st player, and two - against the 2nd. The two strongest players will win 46 games each, while the 3rd and 4th players scored 48 and 47 points, respectively.