The values of points 5, 4, 3, 2, and 1 must be distributed among the three teams. The original question: Was each team awarded at least 1 point?

1) We know that no team was awarded more than a total of 6 points, so any two teams combined weren't awarded more than a total of 12 points. Since the sum of all values of points is 15, two teams are not enough to take all the points. Thus, the answer to the original question is a definite Yes. \(\implies\) Sufficient

2) It's possible that one team got the points 5, 3, 1, another team got the points 4 and 2, while the third team got no points. However, it's also possible that one team got the points 5 and 3, another team got the points 4 and 2, and the third team got the point 1. Thus, we can't get a definite answer to the original question. \(\implies\) Insufficient

Answer: A
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Each team had 3 players, and a team got a point only when the participant was in first five.
We have to check if each team was awarded at least 1 point?-- Since there are 5 positions, we can have 3, 1,1 or 3, 2, 0 or 2, 2, 1 So, two teams will surely get some points, and if know that NO team had more than 2 players in first 5, we can answer YES as the teams would have 2, 2, 1 combination
If we talk of points - TOTAL points = 5+4+3+2+1=15

(1) No team was awarded more than a total of 6 points.
Total points are 15. Ways in which a team would get 6
(a) Various possibility of (2, 2, 1) players--( 5 and 1), (4 and 2), (2) OR (5), (4 and 2), (1 and 3)-- Ofcourse 2, 2, 1 will ensure points to all 3 teams
(b) Only one way where combination is (3, 2, 1) players --(1+2+3),(4),(5), but here too the points are distributed within all three teams
Suff

(2) No pair of teammates finished in consecutive places among the top five places.
If one team got 1st, 3rd and 5th position and other 2nd and 4th--- ans is NO as 3rd team gets 0 point
If tyhe first team got 1st position, second team got 2nd and 4t, and third team got 3rd and 5th---ans is YES
Insuff

A
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The possible number of points are 5,4,3,2,1. We need to see if these points can be spilt between just 2 of the teams or not

A) To take out the worst case scenario, only 2 teams hogging up all the points we will try and fit these points into 2 teams only. Team A takes 3,2,1. Team B will have to take one of 4 or 5 as both combined is more than 6. Hence it is not possible to fit these scores in only 2 teams. All 3 teams will be needed. Sufficient.

B) It is possible to fit in 3 teams. A: 5,3 . B: 4,2. C: 1
and it is also possible to fit in just 2 teams. A: 5,3,1 B: 4,2
Insufficient

A

chetan2u,
Hello Sir, please help me to understand what is the meaning of 6 – n points. how do you come up that the total points are 15. f there are 3 teams with 3 runners each, then why wont the total point is 9.

Hi,

There are 5 positions that will give points..
If some one comes 1, that is n=1, points are 6-1=5..
some one comes 2nd, the points are 6-2=4 and so on 3, 2, 1..
total points = 5+4+3+2+1=15

Hope it helps
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positions: 1_2_3_4_5_6_7_8_9
points (6-n) : 5_4_3_2_1_0_0_0_0

because 1 ≤ n ≤ 5

Statement 1: no team was awarded more than 6 points.
:. points a team could earn 0 ≤ sum of points ≤ 6

Possible combination of points(column wise)
1+5+0.......5+0+0......2+4+0......1+3+0
2+3+0.......4+1+0......1+5+0......4+2+0
4+0+0.......2+3+0......3+0+0......5+0+0

Combinations that will not work:
0+0+0......1+0+0.......1+2+0
1+0+5......2+3+0.......5+0+0
4+3+2......4+5+0.......3+4+0 (all total more than 6 points)

Pretty sure there are more, but I guess the point is there is no way a team could have got 0+0+0 points and the other teams would have 6 or less points. Hence, minimum points a team must have got is 1, given there were no ties.

The 9 participants in a race were divided into 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 1 point?

My approach --

We have 3 teams with 3 participants per team:

Team 1: A, B, C
Team 2: D, E, F
Team 3: G, H, I

Each participant receives 6-n points for achieving nth place. For example: 5 points for achieving 1st place.

Was each team awarded at least one point?

(1) No team was awarded more than a total of 6 points.

The 1st 5 people will receive a certain amount of points. (5+4+3+2+1 = 15 points)
If no team was awarded more than 6 points, we can maximize two teams by awarding them each 6 points. We're then left with 3 points, which must go to the 3rd team. SUFFICIENT.

(2) No pair of teammates finished in consecutive places among the top five places.

1ST: Team 1
2ND: Team 2
3RD: Team 1
4TH: Team 2
5TH: Team 1

In this situation, Team 3 did not receive any points.

1ST: Team 1
2ND: Team 2
3RD: Team 3
4TH: Team 1
5TH: Team 2

In this situation each team received at least one point. INSUFFICIENT.

Answer is A.
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possible points include 1,2,3,4,5

1) If none got > 6, but with 5 spots and 3 teams, 2 teams must have got 2 each (since more than 2 spots violates max 6 points condition). So 1-5, 2-4 and 3 is the only combination in which the spots could have been distributed, indicating that all teams got at least 1 spot. SUFF

2) let the team mates be ABC, DEF, GHI
It is entirely possible that A -> D -> B -> E -> C (leaving out the third team completely)
Hence INSUFF