A8. Twenty students have taken a test. No two students have answered the same number of questions correctly. Each question was answered correctly by at most three students. What is the smallest number of questions the test can have?
A) 63
B) 64
C) 67
D) 70
E) 71

## B-questions
A8. B) 64 Because each of the twenty students had a different number of questions correct, a total of at least $0+1+2+\cdots+19=190$ questions were answered correctly. Since each question was answered correctly by at most three students, there were at least $\frac{190}{3}=63 \frac{1}{3}$ questions, and thus also at least 64.

A situation with 64 questions is also actually possible. Let the twenty students have answered 0 to 19 questions correctly, respectively. We divide the students into three groups:

I consists of the students with 0, 1, 2, 3, 4, 17, 18, or 19 questions correct,

II consists of the students with 5, 6, 7, 14, 15, or 16 questions correct, and

III consists of the students with 8, 9, 10, 11, 12, or 13 questions correct.

For each of the three groups, the total number of correctly answered questions is no more than 64. For each group, we can choose it so that the correctly answered questions of those students are all different. Thus, each question is answered correctly by at most three students.