sukhman wrote:The participants in a race consisted of 3 teams with 3 runners on each team. A team was awarded 6 - n points if one of its runners finished in nth place, where 1 ≤ n ≤ 5. If all of the runners finished the race and if there were no ties, was each team awarded at least one point?
(1) No team was awarded more than a total of 6 points.
(2) No pair of teammates finished in consecutive places among the top five places.
Statement 1:
See whether there is any way to fill the first five places with two teams, each of which got 6 or fewer points.
The total points for the first five places are the following.
1: 6 - 1 = 5
2: 6 - 2 = 4
3: 6 - 3 = 3
4: 6 - 4 = 2
5: 6 - 5 = 1
Total Points: 15
6 + 6 = 12, which is 3 fewer than 15. So there is no way to distribute 15 points to only two teams such that each of two got 6 or fewer points. So all three teams must have had at least one person in the top five places.
Sufficient.
Statement 2:
It is possible to have members of either two or three teams fill the first five places without a pair of members of the same team finishing in consecutive places.
Let the members of the three teams be A1 A2 A3, B1 B2 B3, and C1 C2 C3.
They could finish in the following orders.
1--2--3--4--5
A1 B1 A2 B2 A3 Members of two teams in the first five places.
A1 B1 C1 A2 B2 Members of all three teams in the first five places.
Insufficient.
The correct answer is
A.
