Problem:
Laila took five math tests, each worth a maximum of 100 points. Laila's score on each test was an integer between and , inclusive. Laila received the same score on the first four tests, and she received a higher score on the last test. Her average score on the five tests was . How many values are possible for Laila's score on the last test?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Since the average score on the five tests is , the total score of those five tests must be . Now, let be the score on the first tests, and let be the score for the last test. We know that and , and since , we know Also, since , and dividing by gives us a remainder of , we know that dividing by must leave a remainder of , because will leave no remainder when divided by . Equivalently: .
Since and , the only options for are , and . This gives us four distinct solutions as follows:
Therefore, there are solutions, and is the correct answer.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions