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

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

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

Pls Hit Kudos if you like the solution

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.

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.

there are several teams and each team send 3 participants
Alex scored high than bob... rank of Alex greater than bob(means 18,17,16.....) and score of Alex = medium of all the scores of participants
the rank of bob= 19
the rank of Cathy = 28

we can solve this by evaluating answer choices one by one

A. 6
if there were 6 teams then the total number of
participants would be = 6*3 = 18 (if there are 18 participants how could Cathy get 28th rank so the number of participants must be greater than 18)
this answer choice is wrong

B. 9
similarly total participants = 9*3=27 not possible

C. 10
total participants 10*3= 30
the medium of the scores of 30 participants = \(\frac{15th rank score +16th rank score}{2}\) = Alex score (this cannot be possible since all participants scores are different so alex cant get the medium score in this case)
30 teams are not possible

D. 11
total participants = 11*3 = 33
the medium score of participant = 17th rank students score = Alex score (since Alex got rank >bob and have high score than bob)
this case is possible so correct answer

E. 13
total participants = 13*3= 39
so medium of the score of 39 participants = 20th rank student score
but given that score of Alex = medium of scores of all participants
and alex rank is less than 19 ( that is 18, 17,16....)so this answer choice is not possible

Ranking of Cathy is 28th, hence total number of participants must be 30 or greater multiple of 3.

Also, the score of A is median and everyone scores distinct number. Hence number of participants are odd integer.
Ranking of B is 19 and is greater than A. Hence total number of participants are less than 2*19-1=37

Only possible value of total number of participants is 33 and number of teams=33/3=11.

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?

Total participants = 3 * Total number of teams
Since Bob scored lower than Alex and was ranked 19th, Rank of Alex could be 18, 17, 16....
Alex score equal median score of all participants and all participants scored differently, therefore Alex rank will also be median.
If Alex rank = 18, then total participants = 18+17 = 35 (Not a multiple of 3)
If Alex rank = 17, then total participants = 17+16 = 33 (Multiple of 3 and hence total teams = 11)
If Alex rank = 16, then total participants = 16+15 = 31 (Not a multiple of 3)
If Alex rank = 15, then total participants = 29 (Not a multiple of 3)
If 14, then 27 --> not possible because cathy was ranked 28th.

Option D - 11 is the right answer

A. 6
B. 9
C. 10
D. 11
E. 13

How can one calculate median ?

1 - Arrange the elements of the data set in an ascending or descending order without skipping any element.
2 - Check if the number of elements in the set are ODD or EVEN.
3 - If ODD, Median = the middle element of the arranged set.
If EVEN, Median = Average of the 2 middle elements.

Things we know here -

1. Each team has 3 participants in the test.
2. Alex's score is the median score of all participants.
3. Bob's score < Alex's score.
4. Bob's rank amongst all participants is 19.
5. Cathy's rank amongst all participants is 28

Now, let us look at the options.

Option 1: 6

Number of teams: 6
Number of participants: 6*3 = 18

We can reject this option because we know Bob's rank is 19 and Cathy's rank is 28 which are beyond number of participants as per this option.

A = Wrong.

Option 2: 9

Number of teams: 9
Number of participants: 9*3 = 27

We can reject this option because we know Cathy's rank is 28 which is beyond number of participants as per this option.

B = Wrong.

Option 3: 10

Number of teams: 10
Number of participants: 10*3 = 30

Arrangement of scores of 30 participants can be: 14 scores -- 15th score -- 16th score -- 14 more scores

As per formula for median, Alex's score would be an average of the 15th and 16th score but that would increase the number of scores then by 1 making it 31 scores which is not correct as per the option. Hence,

C = Wrong.

Option 4: 11

Number of teams: 11
Number of participants: 11*3 = 33

Arrangement of scores of 33 participants can be: 16 scores -- 16th score -- 16 more scores

This holds true with all the given information. Hence,

D = Correct.

Option 5: 13

Number of teams: 13
Number of participants: 13*3 = 39

Arrangement of scores of 39 participants can be: 19 scores -- 20th score -- 19 more scores

Here, this does not hold true with the information given as we know Bob's rank is 19th and that Alex's score is greater than Bob's but we know Alex's score is the median of all participants and hence is the 20th score. This is an anomaly because as per ranks, Alex's rank should be less than Bob's which is not holding true. Hence,

E = Wrong.

----

Do give Kudos if you liked the explanation

Here you have to consider some restrictions that are given in the statement:

1) Total quantity of participants is a multiple of 3
2) The order of the team that we are given is like this Cathy (28th) < Bob (19th) < Alex (?)
3) Alex got the score equal to the median of all participants and no two participants got the same score

From (3) we can infer that the total quantity is an odd number, because if it were even, the median would be the average of 2 numbers and could not be Alex's score

Example:

If we have 3 numbers: 18 19 20, the median is 19 - OK
But with 6 numbers: 18 19(alex) 20 21, the median is 19.5, so it differs from the condition of the statement (The median must be Alex's score)

So, after these restrictions, we get that the only possibilites are 11 or 13:

For 13 the distribution of positions would be like this:

19(after alex) 1(alex) 19(before alex)

Discard this one, since with this arrangement, Bob (19th position) would have a better score than Alex

For 11 the distribution of positions would be like this:

16(after alex) 1(alex) 16(before alex)

Comply with all restrictions, so (D) is our answer.

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?

Important things to note before we see the answer choices are:-
1. Each team has sent 3 participants, so the total number of participants will be a multiple of 3.
2. Alex is the median is the set of all participants.
3. Bob is below the median.
4. Cathy is 28th which means that the total number of participants has to be a minimum of 30 or 33 .. since the multiple of 3.

Now let's see the answer choices.

A. 6 (6*3 = 18 which is incorrect as a minimum has to be 30 or 33)

B. 9 (9*3 = 27 which is incorrect as a minimum has to be 30 or 33)

C. 10(10*3 = 30 which matches 1 criterion but the median of 30 is 15.5 which means that 15 & 16 positions would have the same score, which according to the statement is not possible. Everybody has a unique score hence, this is incorrect.)

D. 11 (This is correct as it fits all the above-mentioned criteria. Total participants are 11*3=33)

E. 13 (13*3 = 39 which is incorrect because the median of 39 is 19 which is Bob's position but it is lower than the median)

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?

given :
Alex score is equal to median score
Bob lower score than Alex , rank = 19th
Cathy rank = 28

so Alex scored more than Bob ===> Alex rank better/lower than Bob
so the rank line is expected to look like this ----------------------------> Alex(median) , Bob ,Cathy
-----------------------------------------------------------------------------> …….,Alex(median) ,19,...,28,..
==> Alex rank will be alteast 18 or less , since Alex is also median score he cannot be 18 a even number .
==> Alex should be ranked < 18 and a odd number and
Total team members = Alex rank *2 + 1 > Cathy's rank

Eliminate option A and B as they are too small number of a team , 6*3< Cathy's rank ,9*3< Cathy's rank

Eliminate option C as 3*10 ==>Total members = 30 is a even number , median will be mean of 2 term but not term by itself

Eliminate option E as 13*3 --> Total members = 39 and 19,Bob will be mean here , mean is given as Alex

Left out option is only D , 11*3 =33 > Cathy's rank , Mean is 16 < Bob's rank---Perfectly fits -- Correct

Question: 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?

Of the three participants from NY...
Alex = Median overall
Bob = 19th overall & lower than Alex
Cathy = 28th overall

Total number of participants per team = 3
Total Teams = ?

Putting this information in order gives the following possibilities:
\[
\begin{matrix}
\text{Test #} & \text{# Positions up to Median} & \text{Alex = Median} & \text{Bob} & \text{Cathy} & \text{Total Participants} & \text{Divisible by 3?} \\
1\to & 17 & 18th & 19th & 28th & (17+18 = 35) & 35/3 = No \\
2\to & 16 & 17th & 19th & 28th & (16+17 = 33) & 33/3 = Yes \\
3\to & 15 & 16th & 19th & 28th & (15+16 = 31) & 31/3 = No \\
4\to & 14 & 15th & 19th & 28th & (14+15 = 29) & 29/3 = No \\
5\to & 13 & 14th & 19th & 28th & (13+14 = 27) & 27/3 = N/A \\
\end{matrix}
\]
Test #2 is the only test that gives a total number of participants high enough to include Cathy's position, and that has the ability to be divided by 3 (total number of participants per team).

\(\text{Correct Answer: D} \implies 11 \implies \text{(Total Teams)}\)

----ALEX----|----BOB----|----CATHY----|
--MEDIAN--|----19th----|-----28th----|

Let us see what position can Alex take

• 18th--Total participants(which have to be of form 3*n) will be 17+17+1=35. Not of form 3*n
• 17th--Total participants(which have to be of form 3*n) will be 16+16+1=33. POSSIBLE option
• 16th--Total participants(which have to be of form 3*n) will be 15+15+1=31. Not of form 3*n
• 15th--Total participants(which have to be of form 3*n) will be 14+14+1=29. Not of form 3*n
• 14th--Total participants(which have to be of form 3*n) will be 13+13+1=27. Although of form 2*n, this means total participants is 27, but we know that CATHY has a 28th rank. hence 14th or lower rank for ALEX is not possible.

Hence, IMO the number of total participants is 33. And hence total number of teams = \(33/3=11\) =11
Hence, D

How is the median less than 19? If Bob scored lower than Alex (who has the median score), then Bob's score is left of the median score. If Bob's position is 19, then the median is to the right of the 19th position. Hence the median must be greater than 19. The answer should be E.

EDIT: WOW, I'm supposed to order the numbers in reverse order. That's just mean.

Hi georgethomps

Let's analyze E together. But first we need to understand what the stem is saying:

Stem: Among all Alex got the median score, Bob scored less than Alex (19th), Cathy scored the least among her teammates (28th).

If Bob scored less than Alex and if Bob is 19th, then Alex should be above 19th or at least 18th, right? Please, keep this in mind.

Alex got the median score means that - If we list all the scores from the highest to the lowest, Alex's score would be in the middle of the list.
Now, if we had 13 teams as E says, than the number of the participants would be 39. Note that all scores are different.
If we list all 39 scores, then Alex's score would be in the meadle as it is a median. That means that he is 20th person in the list.

The fact that 'Alex is 20th person in the list' will contradict the stem, becuase Alex should be above than 19th (Bob). Thus E is wrong.

If we have 33 participants, then Alex would be 17th in the list. That's higher than 19th (Bob). Thus D is correct.

I hope I was clear

Since there is 3 ppl per team, the total number of ppl must be Divisible by 3

Given all the Scores are different and Alex has the Median Score, it must be the case that there is an ODD Total Number of Participants.

If there was an EVEN number of Total participants, the only way Alex could receive the Median Score is if 2 ppl got the SAME score, which we are told is impossible.


Case 1:

Let Alex Place just above BOB at 18th Place. Since Alex’s Score is the Median Score in an ODD Set of Numbers that are all different, Alex’s Score is placed at the Unique Middle of the scores when they are listed in Ascending Order.

At 18th Place, there would be 17 ppl ahead of Alex.

Bob would come next at 19th (1 person), 8 more ppl would Place from 20th they 27th, and Cathy would Place 28th (1 person).

7 more ppl would Place lower than Cathy such that the 35th person places Last.

In this case, there would be a total of 35 participants (includ. Alex) ——> a Total which is NOT Evenly Divisible into teams of 3. This case is impossible.


Case 2: Alex places 17th

16 ppl would place ahead of Alex.

Again, since every score is different and Alex’s Score = the Unique Median “middle” Number out of an Odd Total of participants ———> there would be 16 ppl that placed LOWER than Alex (including Cathy and BOB).

16 ppl placed lower than Alex + Alex at 17th Place + 16 ppl placed HIGHER than Alex = 33 participants.

These 33 participants can be Evenly Divided into 11 teams of 3

Answer = 11 teams participated.

-D-

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