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

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<math>\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34</math>
 
<math>\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34</math>
  
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== Solution ==
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== Solution 1 ==
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Let <math>a,ar,ar^{2},ar^{3}</math> be the quarterly scores for the Raiders. We know <math>r > 1</math> because the sequence is said to be increasing. We also know that each of <math>a, ar, ar^2, ar^3</math> is an integer. We start by showing that '''<math>r</math> must also be an integer.'''
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Suppose not, and say <math>r = m/n</math> where <math>m>n>1</math>, and <math>\gcd(m,n)=1</math>. Then <math>n, n^2, n^3</math> must all divide <math>a</math> so <math>a=n^3k</math> for some integer <math>k</math>. Then <math>S_R = n^3k + n^2mk + nm^2k + m^3k < 100</math> and we see that even if <math>k=1</math> and <math>n=2</math>, we get <math>m < 4</math>, which means that the only option for <math>r</math> is <math>r=3/2</math>. A quick check shows that even this doesn't work. Thus <math>r</math> must be an integer.
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Let <math>a, a+d, a+2d, a+3d</math> be the quarterly scores for the Wildcats. Let <math>S_W = a+(a+d) + (a+2d)+(a+3d) = 4a+6d</math>. Let <math>S_R = a+ar+ar^2+ar^3 = a(1+r)(1+r^2)</math>. Then <math>S_R<100</math> implies that <math>r<5</math>, so <math>r\in \{2, 3, 4\}</math>. The Raiders win by one point, so<cmath>a(1+r)(1+r^2) = 4a+6d+1.</cmath>
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*If <math>r=4</math> we get <math>85a = 4a+6d+1</math> which means <math>3(27a-2d) = 1</math>, which is absurd.
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*If <math>r=3</math> we get <math>40a = 4a+6d+1</math> which means <math>6(6a-d) = 1</math>, which is also absurd.
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*If <math>r=2</math> we get <math>15a = 4a+6d+1</math> which means <math>11a-6d = 1</math>. Since <math>15a<100</math> we get <math>a<7</math>. Checking for the allowed values of <math>a</math> by hand we get <math>a=5</math> and <math>d=9</math>.
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Then the quarterly scores for the Raiders are <math>5, 10, 20, 40</math>, and those for the Wildcats are <math>5, 14, 23, 32</math>. Also <math>S_R = 75 = S_W + 1</math>. The total number of points scored by the two teams in the first half is <math>5+10+5+14=\boxed{\textbf{(E)}\ 34}</math>.
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== Solution 2 ==
  
 
Let <math>a,ar,ar^{2},ar^{3}</math> 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 <math>a,a+d,a+2d,a+3d</math> be the quarterly scores for the Wildcats. The sum of the Raiders scores is <math>a(1+r+r^{2}+r^{3})</math> and the sum of the Wildcats scores is <math>4a+6d</math>. Now we can narrow our search for the values of <math>a,d</math>, and <math>r</math>. Because points are always measured in positive integers, we can conclude that <math>a</math> and <math>d</math> are positive integers. We can also conclude that <math>r</math> is a positive integer by writing down the equation:
 
Let <math>a,ar,ar^{2},ar^{3}</math> 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 <math>a,a+d,a+2d,a+3d</math> be the quarterly scores for the Wildcats. The sum of the Raiders scores is <math>a(1+r+r^{2}+r^{3})</math> and the sum of the Wildcats scores is <math>4a+6d</math>. Now we can narrow our search for the values of <math>a,d</math>, and <math>r</math>. Because points are always measured in positive integers, we can conclude that <math>a</math> and <math>d</math> are positive integers. We can also conclude that <math>r</math> is a positive integer by writing down the equation:

Revision as of 12:00, 18 December 2019

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 1

Let $a,ar,ar^{2},ar^{3}$ be the quarterly scores for the Raiders. We know $r > 1$ because the sequence is said to be increasing. We also know that each of $a, ar, ar^2, ar^3$ is an integer. We start by showing that $r$ must also be an integer.

Suppose not, and say $r = m/n$ where $m>n>1$, and $\gcd(m,n)=1$. Then $n, n^2, n^3$ must all divide $a$ so $a=n^3k$ for some integer $k$. Then $S_R = n^3k + n^2mk + nm^2k + m^3k < 100$ and we see that even if $k=1$ and $n=2$, we get $m < 4$, which means that the only option for $r$ is $r=3/2$. A quick check shows that even this doesn't work. Thus $r$ must be an integer.

Let $a, a+d, a+2d, a+3d$ be the quarterly scores for the Wildcats. Let $S_W = a+(a+d) + (a+2d)+(a+3d) = 4a+6d$. Let $S_R = a+ar+ar^2+ar^3 = a(1+r)(1+r^2)$. Then $S_R<100$ implies that $r<5$, so $r\in \{2, 3, 4\}$. The Raiders win by one point, so\[a(1+r)(1+r^2) = 4a+6d+1.\]

  • If $r=4$ we get $85a = 4a+6d+1$ which means $3(27a-2d) = 1$, which is absurd.
  • If $r=3$ we get $40a = 4a+6d+1$ which means $6(6a-d) = 1$, which is also absurd.
  • If $r=2$ we get $15a = 4a+6d+1$ which means $11a-6d = 1$. Since $15a<100$ we get $a<7$. Checking for the allowed values of $a$ by hand we get $a=5$ and $d=9$.

Then the quarterly scores for the Raiders are $5, 10, 20, 40$, and those for the Wildcats are $5, 14, 23, 32$. Also $S_R = 75 = S_W + 1$. The total number of points scored by the two teams in the first half is $5+10+5+14=\boxed{\textbf{(E)}\ 34}$.


Solution 2

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 the quarterly scores for the Wildcats. 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:

\[a(1+r+r^{2}+r^{3})=4a+6d+1\]

Now we can start trying out some values of $r$. 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 $14$.

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

See also

2010 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 18 		Followed by
Problem 20
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All AMC 12 Problems and Solutions

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