1999 AMC 12 Problem 14

Question

Four girls — Mary, Alina, Tina, and Hanna — sang songs in a concert as trios, with one girl sitting out each time. Hanna sang

7

songs, which was more than any other girl, and Mary sang

4

songs, which was fewer than any other girl. How many songs did these trios sing?

7

8

9

10

11

Each song has

3

singers, so the total number of girl-appearances is

3N

Alina and Tina each sang

5

or

6,

and the total must be a multiple of

3

If

N

songs are sung, the total number of girl-appearances is

3N.

Alina and Tina each sang strictly between

4

and

7,

so each sang

5

or

6.

Then

3N = 7 + 4 + (\text{Alina}) + (\text{Tina}),

which is

21, 22,

or

23.

Only

21

is a multiple of

3,

so

N = 7.

Thus, the correct answer is A.

Problem 14 in other years