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

[Albert1]

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

 

[Albert1]

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:

Spoiler

with the restriction :
no three students having the same score ,find min(S) and max(S)

 

[lfdahl]

My attempt:

Spoiler

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.

 

[Albert1]

My attempt:

Spoiler

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