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Several teams competed in a mathematical Olympiad. Each team sent thre

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Several teams competed in a mathematical Olympiad. Each team sent thre  [#permalink]

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New post 09 Jul 2019, 08:00
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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

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Several teams competed in a mathematical Olympiad. Each team sent thre  [#permalink]

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New post 09 Jul 2019, 08:14
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Since no two participants got the same score, if Alex's score is the median and the median is in the set of scores, then the total number of scores (or participants) must be odd which means the total number of teams must be odd

Eliminate (A), (B) & (C)

If the total number of teams is 13 then there are a total of 39 participants. Median score will be the 20th score. This means Alex has to be the 20th rank but we know for a fact that Alex is above the 19th rank.

Eliminate (E)

Answer is (D)
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Several teams competed in a mathematical Olympiad. Each team sent thre  [#permalink]

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New post Updated on: 10 Jul 2019, 20:45
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Alex has the median score. And we are told no two participants received same score. So the question indicates that overall participants will be odd.
Now Bob scored lower than Alex. Bob's position is 19th. So median position is will be better than 19th. So position of Alex will be less than 19.
Now we are told Cathy is at 28th position. So total participants will be more than 28. We are told each team has three members. So minimum teams required is 10.
Now lets look at the options:
1. A and B are less than 10 teams - hence they are eliminated.

2. We concluded as above that total number of participants has to be odd. If we have 10 teams. it will not be true. Hence C is eliminated.

3. Option D says we have 11 teams, hence 33 participants. Median will be 17th position which will be better position than 19th (Bob's position). Also total participants are odd and number of participants greater than 28. So it looks Good . - Correct

4. Option E says there are 13 teams so the number of participants will be 39. Hence 20th position partition will be median. In which case Bob who is at 19th position cannot have scored lower than Alex who is at median position - So it is Incorrect

Answer is D

Originally posted by ruchik on 09 Jul 2019, 08:13.
Last edited by ruchik on 10 Jul 2019, 20:45, edited 2 times in total.
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Re: Several teams competed in a mathematical Olympiad. Each team sent thre  [#permalink]

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New post 09 Jul 2019, 08:13
E is the correct answer.
Because of the scores of B>A>C. B is 19, C is 28, hence A has to be between 19-28.

Each team has 3 participants. If 13 teams are there so there are 39 people each with a different score. So the median will be 20, which falls between 19 and 28.
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Re: Several teams competed in a mathematical Olympiad. Each team sent thre  [#permalink]

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New post 09 Jul 2019, 08:21
if no two participants got the same score
and Alex got the score of the median - her position is the median position. Median formula is n+1/2
We know the median's position is less than 19 here.
And we can assume the no. of terms are odd since everyone sent 3 team members

So,
Cathy is 28th position, what would be the median if terms are 29?
29+1/2 = 15th term.

Total teams here would be 29/3 = 13.

Hence answer is E
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Re: Several teams competed in a mathematical Olympiad. Each team sent thre  [#permalink]

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New post 09 Jul 2019, 08:23
IMO Answer is E:

I solved this by looking at the answer choices.
A. 6 teams --> 18 members so, Bob cannot be 19th position - eliminate
B. 9 teams --> 27 members so, C cannot be on 28th position - eliminate
C. 10 teams --> 30 members, so, arranging in ascending order, Alex would be at 15.5 position, but B is lower than Alex, so, not possible

D. 11 teams --> 33 members, same logic as C

E. 13 teams --> only possible answer
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Several teams competed in a mathematical Olympiad. Each team sent thre  [#permalink]

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New post Updated on: 09 Jul 2019, 23:05
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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 and Cathy's was 28.

So, Alex's position < 19.

Cathy's position = 28. => no of teams >9, since there are 3 members per team and 28/3 > 9. Eliminate A and B

Since alex's position is the median.

Since there are no two people with same score and the median is not average of two scores, the median has to be an odd number. This eliminates even number of participants. Eliminate C.

Alex's max position = 18, max total participants = 35 (which is not possible since there are 3 per team) eliminates E.

Lets say Alex's position = 17. Which implies median = 17.
=> Total number of participants = 33. => no of teams = 33/3 = 11.

Option D.

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Originally posted by prashanths on 09 Jul 2019, 08:25.
Last edited by prashanths on 09 Jul 2019, 23:05, edited 2 times in total.
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Re: Several teams competed in a mathematical Olympiad. Each team sent thre  [#permalink]

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New post 09 Jul 2019, 08:27
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Answer is D

Since Alex got the median score and Bob score less than Alex and his position is 19th, that means Alex must be in better position than Bob. Therefore, if Alex is on 18th position then number of players would be,
n+2/2 = 18, which is 34. Since each team has sent 3 members so 34 cannot be the answer.
If the position of Alex is 17th then total number of players would be 33, and the participating team is 11.

Answer is D
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Re: Several teams competed in a mathematical Olympiad. Each team sent thre  [#permalink]

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New post 09 Jul 2019, 08:27
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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

Since Cathy was ranked 28th
and each team sent 3 participants
T>=10 and n>=30 since n is multiple of 3 and >28, T-# of teams, n=#of participants

Let us take n=30 => median = average of 15th and 16th number
But Alex has median score of all participants and no 2 participants got the same score.
n<>30

Let us take n=33 => Median = 17th term = Alex score. Possible

Let us take n=39 => Median = 20th term = Alex score (20th Position). NOT POSSIBLE since Bob was at 19th Position and had lower score than Alex.

One n=33 is possible => no of teams =33/3 = 11

IMO D
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Re: Several teams competed in a mathematical Olympiad. Each team sent thre  [#permalink]

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New post 09 Jul 2019, 08:27
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Since the total members can be multiple of 3.. hence team members will be greater than 28 (rank of Cathy) (which can be 30,33,36,39 and so on).
Alex is the median one hence the number of members are odd. so it can be (33,39 or so on)
Bob's rank is 19 (lower then Alex's) hence Alex rank should be lower than 19 - (33+1=34)/2=17 and not 40/2=20. hence 33 team members and 11 teams with 3 members each.
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Re: Several teams competed in a mathematical Olympiad. Each team sent thre  [#permalink]

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New post 09 Jul 2019, 08:33
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D i.e, 11 teams

If Cathy is 28th and Alex and Bob are participating, we know there are minimum 30 participants or 10 teams.

Also, Bob is 19th stands at lower rank than Alex, who stands at median, so total participants have to be 33, not more or less.
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Re: Several teams competed in a mathematical Olympiad. Each team sent thre  [#permalink]

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New post 09 Jul 2019, 08:35
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Let us have a look at what we are given:
There are N teams in the competition. Each of N teams has 3 members. In the team from NY there are three participants: Alex (A), Bob (B), and Cathy (C). Alex has the median score, Bob is 19th and has a lower score than Alex, and Cathy is 28th. There are no two people with the same score. We need to find out what is N.

As Bob is 19th in the competition and his score is lower than Alex's, it means that Alex would be ranked 18th or higher in the competition.
Also, taking into account that Cathie is 28th, it means that 10 teams at least were taking part in the competition as each team had 3 members in it.
30 / 3 = 10.

As Alex had a median score among all participants, it means that the same number of participants has a score higher and lower than Alex by terms of median. Taking into account that there were at least 30 people in the competition, Alex would get a place from 15 to 18 in the competition. He cannot be any lower than 18 since his teammate Bob got the place lower than him and was 19th.
An important observation is given that there were no any two participants who would get the same score i.e. no one would share his place with other participant.
In case there are 30 participants, as Alex has a median score, his position would be between 15 and 16, but according to the task conditions, it is not possible as there were no two people with the same score.
The next number divided by 3 is 33: In case there are 33 participants Alex would be 17th. It is still lower than 19th, thus the number of teams N=33/3=11.
The option E is not suitable since if there are 13 teams, 13*3=39, and median for 39 members would be 20, but we know that Alex's position was better.

Answer: D
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Several teams competed in a mathematical Olympiad. Each team sent thre  [#permalink]

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New post Updated on: 09 Jul 2019, 10:00
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we can determine answer from the given options ;
negate option A & B ; because total possible ranks in that case would be 6*3 ; 18 and 9*3; 27 since Cathy rank is 28th so total teams have to be atleast 10
now for option C; 10 teams the total candidates ; 10*3 ; 30 given that no two participants got the same score so total score of all teams; 30*31/2 = 465
and median would 465/30 ; 15.5 ; only integer scores would be valid so its wrong
option D; 11; total members ; 11*3 ; 33 and score of all 33 would be 33*34/2 ; 561 and median ; 561/33 ; 17 is the score of Alex and so will be his rank
option E; 13 ; total members ; 13*3 ; 39 and score of all 39 ; 39*40/2 ; 780 and median ; 780/39 ; 20 ; Alex score ; but given that Bob is rank 19th so this is not correct
IMO 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

Originally posted by Archit3110 on 09 Jul 2019, 08:36.
Last edited by Archit3110 on 09 Jul 2019, 10:00, edited 1 time in total.
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Re: Several teams competed in a mathematical Olympiad. Each team sent thre  [#permalink]

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New post 09 Jul 2019, 08:38
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When one member is 28th then minimum 10 teams (10*3 = 30 members) has to be there - Eliminate option A and D

A. 6 - Cathy is 28th - so not possible as stated above
B. 9 - Cathy is 28th - so not possible as stated above
C. 10 - Median will not be an integer i.e. (15+16)/2 - Given Alex was at median position (not possible)
D. 11 - Correct as median will be 17 which matches the requirements
E. 13 Given 19th is lower than median - hence not possible as median for 13 teams will be 19.

IMO D
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Re: Several teams competed in a mathematical Olympiad. Each team sent thre  [#permalink]

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New post 09 Jul 2019, 08:39
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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

In my opinion correct answer is D - 11;

6 and 9 cant be the options as if these are so then participant count will be 18 & 27 respectively. but as Cathy ranked 28, number of participants will be >28 & multiple of 3.

Coming to C; 10 - for this case we have even number of participation i.e, 30, median score will be the avg of middle 2 numbers when the numbers are sorted from smallest to largest - Alex got the median score & i.e when sorted, it will be the avg of middle 2 nos, but as bob's score is<Alex's & Bob's rank is 19, it will not be the case

D - 11, total participant is 33, median number will be when sorted, 17th term & Alex's rank will be 17 & when reversely followed, Bob is at 19 at a lower score.

E - 13 - total participant is 39, median is the avg of middle 2 numbers when the numbers are sorted from smallest to largest & will be greater than Bob.

I am not sure but may be someone can help me understand in a better way.
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Re: Several teams competed in a mathematical Olympiad. Each team sent thre  [#permalink]

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New post 09 Jul 2019, 08:41
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This one was pretty simple as

We know the median will be less than 19

So there can't be offcourse 6,9 teams as total number of players will.be less than 28

Then 39 is not the option too as median will be 20 and this can't be median as median is less than 19

In 11 and 10

Correct answer is 11 as 10eams median is ratio of 15 and 16 and no two score can be the same .

Correct is 11

Option D is correct

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Several teams competed in a mathematical Olympiad. Each team sent thre  [#permalink]

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New post Updated on: 10 Jul 2019, 00:27
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Only D works here.
The number of groups can not be lower than 10 (or 30 participants) because if lower, then question will be a mess.
In A, we have only 18 members (6*3), but C person was 28th. Thus out.
Same with B, if there are 27 members (9*3), then C person is not in the games, since she is ranked 28th. Out.
In C, 10*3 or median is 15.5, not integer. This option does not work because A cannot be ranked as 15.5, he is either 15 or 16. Out.
D works, we have 33 participants (11*3), and median is 17, B is 19th (lower than A) and C is 28th out of 33 members. Let's keep this option.
E, 13*3=39, median should be 20, but we know that B, 17th, got lower score than median did. With E this is not possible. Out.
Only D

Originally posted by mira93 on 09 Jul 2019, 08:42.
Last edited by mira93 on 10 Jul 2019, 00:27, edited 1 time in total.
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Several teams competed in a mathematical Olympiad. Each team sent thre  [#permalink]

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New post Updated on: 09 Jul 2019, 09:09
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Rank of Alex = Median
Rank of Bob = 19 (Less than Alex)
Rank of Cathy = 28 (Least ranked among 3)

So ranking order = A > B > C

Possible Ranking of A = 18, 17, 16 . . . etc
Total Number of Teams should be ODD because Median has to have a UNIQUE Rank.

A. 6 : Even --> Not Possible

B. 9
Total Players = 27
Rank of Cathy = 28 --> Not Possible

C. 10 : Even --> Not Possible

D. 11
Total Players = 33
Median Rank = 17 --> Possible

E. 13
Total Players = 39
Median Rank = 20
Rank of Bob = 19 -->Not Possible


IMO Option D

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Originally posted by Dillesh4096 on 09 Jul 2019, 08:44.
Last edited by Dillesh4096 on 09 Jul 2019, 09:09, edited 1 time in total.
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Several teams competed in a mathematical Olympiad. Each team sent thre  [#permalink]

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New post 09 Jul 2019, 08:44
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What would be the total number of participants for each different answer?

A. 6 -> 18
B. 9 -> 27

C. 10 -> 30
D. 11 -> 33
E. 13 -> 39

1st Info: "Cathy was ranked at 28th position among all the participants" - There must be at least 28 participants, thus we can eliminate A and B.

2nd Info: "Bob received lower score than Alex and was ranked at 19th position among all the participants" + "Alex got the score equal to the median score". The median is less than 19, thus we can eliminate E because if we had 39 participants, the median would be 19.5, which is higher than 19.

3rd Info: "Alex got the score equal to the median score of all the participants" + "no two participants got the same score". This tells us that the median is an Integer.

- If we had 30 participants, the median would be the score between the 15th and the 16th ranked participant. This is impossible because we are told that Alex is equal to the median and no two participants got the same score. If the 15th and the 16th participant would have the same score, the answer could be C, but it's not the case - Eliminate C.

- If we had 33 participants, the median would be the score of the 17th ranked participant. Last option, 17 is an Integer, everything is ok: (D) 11 teams is the solution.
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Re: Several teams competed in a mathematical Olympiad. Each team sent thre  [#permalink]

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New post 09 Jul 2019, 08:46
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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?

We know that:
1. Median is lower than 19 because Alex (Median) scored better than Bob who is ranked 19th
2. There are atleast 28 players in all because Cathy is ranked 28.
3. The total number of players is ODD, because no player has same score and alex has median score. If it was even, media would be average of 2 different numbers which could not be equal to alexs score.
4. Number of players is divisible by 3 because each team has 3 players.

We need to find number of total players (N). we can then divide N by 3 to get number of teams (T).

Since N is >= 28 we will start there.

28... not possible... because even
29... not possible.... because not divisible by 3
30... not possible.... because even
31... not possible... not divisible by 3
32... not possible... even
33... possible... all conditions satisfied... if N = 33, T = 11 ... so options A, B, and C are eliminated.

We will check option E just to double check... if T = 13, N = 39... if N = 39, median would be 19... which is not possible because median is lower than 19.... so D is also eliminated.

Answer D - 11.
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