In a chess tournament, 8 players participated, each playing once against every other participant. Each player earned a different number of points. The player who won the second prize earned as many points as the last four placed players combined. What was the result of the match between the 3rd and 7th place players?
I. solution. Each player played 7 matches, so the first place could have a maximum of 7 points (as is customary, each match counts for 1 point), and the second place could have a maximum of 6. If the second place had 6.5 points, that is, only a 0.5 point loss compared to the possibility, then they would have scored at least 0.5 points in their match against the first place, and thus the first place could not have achieved 7 points; their point totals would be equal, which contradicts the conditions of the problem.

On the other hand, the second place had at least 6 points, as it was the same as the total points of the 5th to 8th places, and among the latter, the 6 points from the 6 matches played among themselves (5th-6th, 5th-7th, 5th-8th, 6th-7th, 6th-8th, 7th-8th) are certainly included. Therefore, the point total of the second place and the combined point total of the 5th to 8th places are exactly 6, so the latter did not score a single point against the 1st to 4th places, which means the 3rd place won the match in question.

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\text { László Németh (Győr, Benedek-rendi Czuczor G. g. II. o. t.) }
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II. solution. We saw in the first solution that the second player could have a maximum of 6 points. We will show that it cannot be less. The total points are 28 (since the total points are the same regardless of how each match ends; if, for example, every match ends in a draw, then each player's point total is $7 \cdot 0.5 = 3.5$, and the total points are 8 times that). If the second player had only 5.5 points, then the 5th to 8th players together would also have 5.5 points, and the third would have a maximum of 5, and the fourth a maximum of 4.5 points. Together, this would be a maximum of 20.5 points, so the first would have at least 7.5 points, which is not possible.

Let's see if and how many points the third player could have lost to the seventh. The first player has 6.5 or 7 points, the second has 6, and the 5th to 8th together have 6, so the third and fourth together have 9.5 or 9 points. More points go to the third than to the fourth, so the third has at least 5 points, and at most 5.5 points. Thus, the third could end with a maximum of 2 points lost, and if they lost at least 0.5 points to the seventh, they could have lost a maximum of 1.5 more points, and they had to win at least 0.5 points against the first and second.

The first either won or drew against the second. In the former case, the second won against everyone else, so the third could only win the mentioned at least 0.5 points from the first (and only in a draw), and the fourth had to lose to the first, the second, and the third. In the latter case, the first could not lose more points, so the third could only draw with the second. In this case, too, all three had to win against the fourth, as they could not lose more points.

In both cases, we found that the first has 6.5 points, the second has 6, the 5th to 8th have the same, the third has 5, and the fourth has at most 4. This would be a maximum of 27.5 points together, which is not possible. Therefore, the third place must have won against the seventh (and it also follows that the 5th to 8th place players each lost to the third).

Note. Several people examined what possibilities there are for the second player to lose 1 point. This required them to try many more possibilities, but it also turned out that the third place must have won the match against the seventh in every case.