Question

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?

$30$

$31$

$32$

$33$

$34$

Solution:

Let the Raiders score $a, ar, ar^2, ar^3$ (increasing geometric, $r\gt1$ ) and the Wildcats $a, a+d, a+2d, a+3d$ (increasing arithmetic), tied in the first quarter at $a.$

Every quarter score is a positive integer and each total is under $100,$ so the ratio and first term are small. Testing $r=2$ gives Raiders $5, 10, 20, 40$ with total $75.$

The Wildcats then total $74=4a+6d=20+6d,$ so $d=9,$ giving $5, 14, 23, 32.$ The Raiders won $75$ to $74.$

The first-half total is $(5+10)+(5+14)=34.$

Thus, the correct answer is E.

2010 AMC 12B Problem 19

Solution:

Problem 19 in Other Years