5. The final score in figure skating is determined as follows. The judging panel consists of ten people. Each judge gives the athlete their own score for the performance. After that, seven of the ten received scores are randomly selected. The sum of these seven scores is the final score. Places among the athletes are distributed according to the final score obtained: the higher the score, the better the result.

In the championship, 6 athletes participated. Could it have been the case that:

a) the athlete with the maximum sum of all 10 scores came in last place?

b) the athlete with the maximum sum of all 10 scores came in last place, while the athlete with the minimum sum of all 10 scores came in first place?
Solution.

a) a special case of point b).

Answer: Yes, it could.

b) Let's provide an example where such a situation is possible.

| Athlete | Scores |  |  |  |  |  |  |  |  |  | Sum of 10 | Sum of 7 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 25 | 7 |
| 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 5 | 5 | 5 | 23 | 8 |
| 3 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 4 | 4 | 21 | 9 |
| 4 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 3 | 3 | 19 | 10 |
| 5 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | 2 | 2 | 2 | 17 | 11 |
| 6 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 1 | 1 | 1 | 15 | 12 |

In this voting scenario, if only the scores of the first 7 judges are considered, the 1st athlete will take the last place, while the sum of all 10 scores is the highest. At the same time, the 6th athlete will take the first place, and the sum of all 10 scores is the lowest.

Answer: Yes, it could.