8) Michael, Juan Pablo, and Kimi participate in a car racing championship. After 5 grand prix, Michael leads the standings with 43 points, followed by Juan Pablo with 42 points, and Kimi with 40. In each of the 5 races, the first-place finisher earned 10 points, the second-place finisher 8 points, the third-place finisher 7 points, and no points were awarded from fourth place onwards. Based on this information, can you determine who finished in second place the most number of times?
(A) Michael
(B) Juan Pablo
(C) Kimi
(D) Michael and Juan Pablo
(E) Michael and Kimi.
8) The answer is $\mathbf{( E )}$. The total points of Michael, Juan Pablo, and Kimi is 125. Since there are 25 points available in each race, they have taken all the available points, and thus they must have placed first, second, and third in every race. Compared to the maximum possible 50 points, Michael lost 7 points. He could lose 2 points (by placing second) or 3 points (by placing third), so the only possibility is that he placed second twice and third once. Juan Pablo, who lost 8 points compared to the maximum possible, could have placed third twice and second once, or second four times. However, the latter possibility is excluded, since Michael has collected two second places, and no one else can have placed second more than three times. By elimination, Kimi must have placed second twice and third twice.

SECOND SOLUTION.

A simpler solution can be found by considering the remainder of the scores of the three competitors when divided by 3: the scores for first and third place have a remainder of 1, and the score for second place has a remainder of 2. Therefore, it can be seen quickly that:

| placements | remainder of the score when divided by 3 |
| :--- | :--- |
| every time first or third | $1+1+1+1+1=5$, remainder 2 |
| once second | $2+1+1+1+1=6$, remainder 0 |
| twice second | $2+2+1+1+1=7$, remainder 1 |
| three times second | $2+2+2+1+1=8$, remainder 2 |
| four times second | $2+2+2+2+1=9$, remainder 0 |
| five times second | $2+2+2+2+2=10$, remainder 1 |

Moreover, Michael's score has a remainder of 1, Juan Pablo's score has a remainder of 0, and Kimi's score has a remainder of 1. Therefore, Michael and Kimi must have qualified second twice each, and Juan Pablo only once. The correct answer is then (E). Indeed, the placements could have been as follows:

| Race | 1 | 2 | 3 | 4 | 5 |
| :---: | :---: | :---: | :---: | :---: | :---: |
| Michael | 10 | 10 | 8 | 8 | 7 |
| Juan Pablo | 7 | 7 | 10 | 10 | 8 |
| Kimi | 8 | 8 | 7 | 7 | 10 |