Difference between revisions of "2010 AMC 12B Problems/Problem 19"

Revision as of 18:00, 31 May 2011

Problem 19

A high school basketball game between the Raiders and 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?

$\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34$

Solution

Let $a,ar,ar^{2},ar^{3}$ be the quarterly scores for the Raiders. We know that The Raiders and Wildcats both scored the same number of points in the first quarter so let $a,a+d,a+2d,a+3d$ be quarterly scores for the Raiders. The sum of the Raiders scores is $a(1+r+r^{2}+r^{3})$ and the sum of the Wildcats scores is $4a+6d$ . Now we can narrow our search for the values of $a,d$ , and $r$ . Because points are always measured in positive integers, we can conclude that $a$ and $d$ are positive integers. We can also conclude that $r$ is a positive integer by writing down the equation:

$a(1+r+r^{2}+r^{3})=4a+6d+1$

Because both the left hand and right hand sides must be integers, we have that $r+r^{2}+r^{3}$ is a positive integer and due to our restrictions, it follows that $r$ must be a positive integer as well. Now we can start trying out some values of $r$ . We start at $r=1$ which gives that $6d=-1$ which is a contradiction. Next we try $r=2$ , which gives

$15a=4a+6d+1$ $11a=6d+1$

We need the smallest multiple of $11$ (to satisfy the <100 condition) that is $\equiv 1 \pmod{6}$ . We see that this is $55$ , and therefore $a=5$ and $d=9$ .

So the Raiders first two scores were $5$ and $10$ and the Wildcats first two scores were $5$ and $14$ .

$5+10+5+14=34 \longrightarrow \boxed{\textbf{(E)}}$

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 == Problem 19 ==
 A high school basketball game between the Raiders and 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 <math>100</math> points. What was the total number of points scored by the two teams in the first half?

2010 AMC 12B (Problems • Answer Key • Resources)
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Difference between revisions of "2010 AMC 12B Problems/Problem 19"

Revision as of 18:00, 31 May 2011

Problem 19

Solution

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