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30 Nov 2021, 07:47
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Several teams competed in a mathematical Olympiad. Each team sent three participants. Alex, Bob, and Cathy were the three members of the team from NY. Alex got the score equal to the median score of all the participants, Bob received lower score than Alex and was ranked at 19th position among all the participants, and Cathy was ranked at 28th position among all the participants. How many teams took part in the Olympiad if no two participants got the same score?
A. \(6\)
B. \(9\)
C. \(10\)
D. \(11\)
E. \(13\)
A. \(6\)
B. \(9\)
C. \(10\)
D. \(11\)
E. \(13\)
30 Nov 2021, 07:47
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Official Solution:
Several teams competed in a mathematical Olympiad. Each team sent three participants. Alex, Bob, and Cathy were the three members of the team from NY. Alex got the score equal to the median score of all the participants, Bob received lower score than Alex and was ranked at 19th position among all the participants, and Cathy was ranked at 28th position among all the participants. How many teams took part in the Olympiad if no two participants got the same score?
A. \(6\)
B. \(9\)
C. \(10\)
D. \(11\)
E. \(13\)
1. Since, no two participants got the same score, then the fact that Alex got the score equal to the median score means that the total number of participants must odd (Alex being at odd position).
2. Since Cathy was ranked at 28th position, then the total number of participants must be a multiple of 3 (because each team sent three participants) which is more than or equal to 30: 30, 33, 36, ...
3. Since Bob received lower score than Alex and was ranked at 19th, then Alex could be ranked as low as at 17th position, making the maximum number of participants possible equal to \(16*2+1=33\) (16 participants at lower positions than Alex, 16 participants at higher positions than Alex, and Alex herself, at 17th position).
So, we are looking for an odd multiple of 3 between 30 and 33, inclusive. The only possible value is 33. The number of teams is therefore, \(\frac{33}{3}=11\).
Answer: D
Several teams competed in a mathematical Olympiad. Each team sent three participants. Alex, Bob, and Cathy were the three members of the team from NY. Alex got the score equal to the median score of all the participants, Bob received lower score than Alex and was ranked at 19th position among all the participants, and Cathy was ranked at 28th position among all the participants. How many teams took part in the Olympiad if no two participants got the same score?
A. \(6\)
B. \(9\)
C. \(10\)
D. \(11\)
E. \(13\)
1. Since, no two participants got the same score, then the fact that Alex got the score equal to the median score means that the total number of participants must odd (Alex being at odd position).
2. Since Cathy was ranked at 28th position, then the total number of participants must be a multiple of 3 (because each team sent three participants) which is more than or equal to 30: 30, 33, 36, ...
3. Since Bob received lower score than Alex and was ranked at 19th, then Alex could be ranked as low as at 17th position, making the maximum number of participants possible equal to \(16*2+1=33\) (16 participants at lower positions than Alex, 16 participants at higher positions than Alex, and Alex herself, at 17th position).
So, we are looking for an odd multiple of 3 between 30 and 33, inclusive. The only possible value is 33. The number of teams is therefore, \(\frac{33}{3}=11\).
Answer: D
08 Jan 2025, 05:46
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The answer wont be E)13 because 13*3 = 39. The median value would be 19. Alex and Bob would get same rank which is not possible. So option D) 11 fills the criteria
