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

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== Problem 19 ==
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{{duplicate|[[2010 AMC 12B Problems|2010 AMC 12B #19]] and [[2010 AMC 10B Problems|2010 AMC 10B #24]]}}
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== Problem ==
 
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?
 
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?
  
 
<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 not possible with the given conditions.
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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 not possible with the given conditions.
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*If <math>r=2</math> we get <math>15a = 4a+6d+1</math> which means <math>11a-6d = 1</math>. Reducing modulo 6 we get <math>a \equiv 5\pmod{6}</math>. Since <math>15a<100</math> we get <math>a<7</math>. Thus <math>a=5</math>. It then follows that <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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Note if you don't realize while taking the test that <math>r</math> might not be an integer: since an answer is achieved through casework on the integer value of <math>r</math> and since there is only one right answer, the proof of <math>r</math> being an integer can be skipped on the test (it takes up time).
  
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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:
  
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<cmath>5+10+5+14=34 \longrightarrow \boxed{\textbf{(E)}}</cmath>
 
<cmath>5+10+5+14=34 \longrightarrow \boxed{\textbf{(E)}}</cmath>
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== Solution 3 (Quick Solve?) ==
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It should become apparent that the geometric ratio is less than or equal to <math>2</math> (try the first quarter <math>1</math>, ratio <math>3</math>). Obviously, the ratio is more than <math>1</math>, so we try the ratio of <math>2</math>
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<math>1, 2, 4, 8</math> - The Raiders
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<math>1, ?, ?, ?</math> - The Wildcats
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To find the <math>?</math>s, we need the sum of the raiders, which is <math>15</math>. <math>15-1=14</math>, and <math>\frac{14}2=7</math>. So the last term is <math>7-1=6</math>. But, now we can't have integrated 2nd and 3rd quarter scores, so we stop.
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<math>2, 4, 8, 16</math> - The Raiders, this also fails (try it out)
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The same goes for <math>3, 6, 12, 24</math>, and <math>4, 8, 16, 32</math>. With <math>5, 10, 20, 40</math> we get:
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<math>5, 10, 20, 40</math> - The Raiders
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<math>5, ?, ?, ?</math> - The wildcats
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<math>5+10+20+40=75</math>, <math>75-1=74</math>, <math>\frac{74}2=37</math>, <math>37-5=32</math>
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<math>5, ?, ?, 32</math> - The Wildcats.
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Now, we can finish the arithmetic progression:
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<math>5, 14, 23, 32</math> - The Wildcats
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Thus our answer is <math>5+10+5+14=34</math>
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~ firebolt360
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== Video Solution ==
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https://youtu.be/krRrPxRdgD0
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~IceMatrix
  
 
== See also ==
 
== See also ==
 
{{AMC12 box|year=2010|num-b=18|num-a=20|ab=B}}
 
{{AMC12 box|year=2010|num-b=18|num-a=20|ab=B}}
 +
{{AMC10 box|year=2010|num-b=23|num-a=25|ab=B}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 18:14, 25 March 2024

The following problem is from both the 2010 AMC 12B #19 and 2010 AMC 10B #24, so both problems redirect to this page.

Problem

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 not possible with the given conditions.
  • If $r=3$ we get $40a = 4a+6d+1$ which means $6(6a-d) = 1$, which is also not possible with the given conditions.
  • If $r=2$ we get $15a = 4a+6d+1$ which means $11a-6d = 1$. Reducing modulo 6 we get $a \equiv 5\pmod{6}$. Since $15a<100$ we get $a<7$. Thus $a=5$. It then follows that $d=9$.

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

Note if you don't realize while taking the test that $r$ might not be an integer: since an answer is achieved through casework on the integer value of $r$ and since there is only one right answer, the proof of $r$ being an integer can be skipped on the test (it takes up time).

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 $14$.

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

Solution 3 (Quick Solve?)

It should become apparent that the geometric ratio is less than or equal to $2$ (try the first quarter $1$, ratio $3$). Obviously, the ratio is more than $1$, so we try the ratio of $2$

$1, 2, 4, 8$ - The Raiders

$1, ?, ?, ?$ - The Wildcats

To find the $?$s, we need the sum of the raiders, which is $15$. $15-1=14$, and $\frac{14}2=7$. So the last term is $7-1=6$. But, now we can't have integrated 2nd and 3rd quarter scores, so we stop.

$2, 4, 8, 16$ - The Raiders, this also fails (try it out)

The same goes for $3, 6, 12, 24$, and $4, 8, 16, 32$. With $5, 10, 20, 40$ we get:

$5, 10, 20, 40$ - The Raiders

$5, ?, ?, ?$ - The wildcats

$5+10+20+40=75$, $75-1=74$, $\frac{74}2=37$, $37-5=32$

$5, ?, ?, 32$ - The Wildcats.

Now, we can finish the arithmetic progression:

$5, 14, 23, 32$ - The Wildcats

Thus our answer is $5+10+5+14=34$

~ firebolt360

Video Solution

https://youtu.be/krRrPxRdgD0

~IceMatrix

See also

2010 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 18 		Followed by
Problem 20
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions
2010 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 23 		Followed by
Problem 25
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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