Problem:
A high school basketball game between the Raiders and the Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than 100 points. What was the total number of points scored by the two teams in the first half?
Answer Choices:
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B.
C.
D.
E.
Solution:
The Raiders' score was , where is a positive integer and . Because is also an integer, for relatively prime positive integers and with . Moreover is an integer, so divides . Let . Then the Raiders' score was ), and the Wildcats' score was for some positive integer . Because , the condition implies that and . The only possibilities are , or . The corresponding values of are, respectively, , and . In the first two cases , and the corresponding values of are, respectively, and . In neither case is an integer. In the third case which is impossible in integers. In the last case , from which . The only solution in positive integers for which is . Thus , , and the number of points scored in the first half was .
The problems on this page are the property of the MAA's American Mathematics Competitions