If S=9999 prove we can find at least 3 students having the same score

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In summary, the conversation discusses a math test with 201 students and their scores from set A. The questions ask to prove that with a total score of 9999 or 10101, there will be at least 3 students with the same score. It is also mentioned that if the total score is 10000 or 10100 and there are no three students with the same score, there will be 1 student with a score of 100 or 0, and 2 students with scores of 0 or 100.
  • #1
Albert1
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A={0,1,2,3,4,5,--------,99,100}
201 students are attending a math test ,the score of each student can be
found from set A,if S represents the total scores of all 201 students,please answer
the following questions
(1) if S=9999 prove we can find at least 3 students having the same score
(2) if S=10101 prove we can also find at least 3 students having the same score
(3) if S=10000 and it is known no three students having the same score
,then there must have 1 student having score 100,and 2 students with scores 0
(4) if S=10100 and it is known no three students having the same score
,then there must have 1 student having score 0,and 2 students with scores 100
 
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  • #2
Albert said:
A={0,1,2,3,4,5,--------,99,100}
201 students are attending a math test ,the score of each student can be
found from set A,if S represents the total scores of all 201 students,please answer
the following questions
(1) if S=9999 prove we can find at least 3 students having the same score
(2) if S=10101 prove we can also find at least 3 students having the same score
(3) if S=10000 and it is known no three students having the same score
,then there must have 1 student having score 100,and 2 students with scores 0
(4) if S=10100 and it is known no three students having the same score
,then there must have 1 student having score 0,and 2 students with scores 100
hint:
with the restriction :
no three students having the same score ,find min(S) and max(S)
 
  • #3
My attempt:

The task is to assign scores ($0,1,.., ,100$) to $201$ students.

Using the hint from Albert, I proceed as follows:

Given the restriction: No three students have the same score, what is $S_{max}$ and $S_{min}$?

I can assign a score only twice: $2$ x $0$, $2$ x $1$, …, $2$ x $100$. Total sum is $10.100$.

This sum corresponds to exactly $202$ assignments, so we need to subtract just one single score. By taking

the largest and the smallest possible score, we easily find the max/min sum:

Thus:

$S_{min} = 10.100 – 100 = 10.000$ (*)

Subtracting $100$ from the total sum means, we have only one student with score $100$ in the test, and two

students with score $0$ (if not, there would only be $199$ students) – wen $S = S_{min}$. This answers problem (3).$S_{max}= 10.100 – 0 = 10.100$. (**).

Subtracting $0$ from the total sum means, we have only one student with score $0$ in the test and two students

with score $100$, - wen $S = S_{max}$. This answers problem (4).From (*) and (**) we can conclude, that: If $10.000 \le S \le 10.100$ then there are no three students having the same score.

in the group of 201 students. Or the complementary statement: Any sum, $S$, smaller than $S_{min}$

or greater than $S_{max}$ will contain at least a triple score.

This answers problem (1) and (2) In Alberts challenge, and we are done.
 
  • #4
lfdahl said:
My attempt:

The task is to assign scores ($0,1,.., ,100$) to $201$ students.

Using the hint from Albert, I proceed as follows:

Given the restriction: No three students have the same score, what is $S_{max}$ and $S_{min}$?

I can assign a score only twice: $2$ x $0$, $2$ x $1$, …, $2$ x $100$. Total sum is $10.100$.

This sum corresponds to exactly $202$ assignments, so we need to subtract just one single score. By taking

the largest and the smallest possible score, we easily find the max/min sum:

Thus:

$S_{min} = 10.100 – 100 = 10.000$ (*)

Subtracting $100$ from the total sum means, we have only one student with score $100$ in the test, and two

students with score $0$ (if not, there would only be $199$ students) – wen $S = S_{min}$. This answers problem (3).$S_{max}= 10.100 – 0 = 10.100$. (**).

Subtracting $0$ from the total sum means, we have only one student with score $0$ in the test and two students

with score $100$, - wen $S = S_{max}$. This answers problem (4).From (*) and (**) we can conclude, that: If $10.000 \le S \le 10.100$ then there are no three students having the same score.

in the group of 201 students. Or the complementary statement: Any sum, $S$, smaller than $S_{min}$

or greater than $S_{max}$ will contain at least a triple score.

This answers problem (1) and (2) In Alberts challenge, and we are done.
Thanks lfdahl: well done
 

FAQ: If S=9999 prove we can find at least 3 students having the same score

How does S=9999 prove that there are at least 3 students with the same score?

Proof by contradiction: Assume there are less than 3 students with the same score. This means there are at most 2 students with unique scores. However, since S=9999, there are 9999 possible scores. Therefore, it is impossible for only 2 students to have unique scores. This contradicts our assumption, so there must be at least 3 students with the same score.

Is S=9999 a sufficient condition for there to be at least 3 students with the same score?

Yes, S=9999 is a sufficient condition for there to be at least 3 students with the same score. This means that if S=9999, then there must be at least 3 students with the same score. However, it is not a necessary condition, as there could be other circumstances that result in 3 or more students having the same score.

Can you provide an example to illustrate this proof?

Sure, let's say there are 6 students with scores of 100, 99, 98, 97, 96, and 95. This is a total of 6 unique scores, but since S=9999, there are 9999 possible scores. Therefore, it is impossible for only 6 students to cover all 9999 possible scores. This means that there must be at least 3 students with the same score.

Does this proof apply to any number of students or only a specific number?

This proof applies to any number of students. As long as S=9999, there must be at least 3 students with the same score. This holds true regardless of the total number of students in the group.

Can this proof be generalized to any other situations?

Yes, this proof can be generalized to other situations as long as the total number of possible outcomes is greater than the number of unique outcomes. For example, if there are 10 possible outcomes and only 9 unique outcomes, then there must be at least 2 instances of the same outcome. This concept can be applied in various fields such as statistics, probability, and data analysis.

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