## Task 2 - 180812

The result of a class test is known as follows:

- More than 20 and fewer than 40 students participated.
- The arithmetic mean of all grades achieved by the students in this test was 2.3125.
- No student received a grade of "5" in this test.
- The number of "2"s was an odd number and greater than 12.
- The number of "3"s was exactly the same as the number of "2"s.

a) Determine the number of students who participated in this class test!

b) How many of them received a grade of "1"?
a) Since $2.3125=\frac{23125}{10000}=\frac{37}{16}$ and since 37 is coprime to 16, the number of participants must be divisible by 16. The only (natural) number that is a (integer) multiple of 16 and lies between 20 and 40 is 32. Therefore, 32 students were involved in this class test.
b) The sum of the grades is 74, since $\frac{37}{16} \cdot 32=74$.

The number of "2s" is not greater than half the number of all students, so according to the problem, it can only be 13 or 15. If 15 "2s" and thus, according to the problem, also 15 "3s" had been written, the sum of these grades would be 75, in contradiction to the sum of 74 determined above.

Therefore, 13 "2s" and 13 "3s" were written. Consequently, the remaining 6 students received either a "1" or a "4".

Since the arithmetic mean of all grades was less than 2.5, the number of "1s" must have been greater than the number of "4s". This leaves only the following possibilities:

| 1 | 2 | 3 | 4 |
| :---: | :---: | :---: | :---: |
| 6 | 13 | 13 | 0 |
| 5 | 13 | 13 | 1 |
| 4 | 13 | 13 | 2 |

Only the second case results in the given arithmetic mean. Therefore, 5 students received the grade "1" in this test.