$$Given\ that\ 1\le n\le5\ and\ point\ =\ 6-n$$
Max point that can be awarded = 5
The minimum point that can be awarded = 1
Total points that can be awarded = 5 + 4 + 3 + 2 + 1 = 15

Statement 1 => No team was awarded more than a total of 6 points.
With the team having the best performance, getting a max point of 6, the least team will have 3 points
i.e team 1 = 6 points; team 2 = 6 points; team 3 = 3 points
Total = 6 + 6 + 3 = 15 points So, each team was awarded more than one point
Statement 1 is SUFFICIENT

Statement 2 => No pair of teammates finished in consecutive places among the top five places.
This means no team gets consecutive points
The total points gotten by each team are unknown since the point of each team is dependent on the performance and position of each runner in the team and position is not given to estimate the points.
Statement 2 is NOT SUFFICIENT

Since statement 1 alone is SUFFICIENT
Answer = A