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Difference between revisions of "2024 AMC 12B Problems/Problem 10"
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==Problem 6==
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==Problem 10==
  
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The national debt of the United States is on track to reach <math>5\times10^{13}</math> dollars by <math>2023</math>. How many digits does this number of dollars have when written as a numeral in base 5? (The approximation of <math>\log_{10} 5</math> as <math>0.7</math> is sufficient for this problem)
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A list of 9 real numbers consists of <math>1</math>, <math>2.2 </math>, <math>3.2 </math>, <math>5.2 </math>, <math>6.2 </math>, <math>7</math>, as well as <math>x, y,z</math> with <math>x\leq y\leq z</math>. The range of the list is <math>7</math>, and the mean and median are both positive integers. How many ordered triples <math>(x,y,z)</math> are possible?
  
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<math>\textbf{(A) } 18 \qquad\textbf{(B) } 20 \qquad\textbf{(C) } 22 \qquad\textbf{(D) } 24 \qquad\textbf{(E) } 26</math>
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<math>\textbf{(A) }1 \qquad\textbf{(B) }2 \qquad\textbf{(C) }3 \qquad\textbf{(D) }4 \qquad\textbf{(E) \text{infinitely many}}\qquad</math>
  
 
==Solution==
 
==Solution==
  
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The number of digits is just <math>\lceil \log_{5} 5\times 10^{13} \rceil</math>.
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It is easiest to do casework on the range. We have four possible cases:
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Note that
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<math>x<1</math>, <math>z<7</math>
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<cmath>\log_{5} 5\times 10^{13}=1+\frac{13}{\log_{10} 5}</cmath>
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<math>x>1</math>, <math>z<7</math>
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<cmath>\approx 1+\frac{13}{0.7}</cmath>
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<math>x<1</math>, <math>z>7</math>
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<cmath>\approx 19.5</cmath>
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<math>x>1</math>, <math>z>7</math>
  
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Hence, our answer is <math>\fbox{\textbf{(B) } 20}</math>
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These encapsulate all possible values of <math>x</math> and <math>z</math> we can choose, so we're not leaving anything out. As we'll see, the problem will become simpler this way. Since we want integer mean, first note the sum of the values given to be <math>24.8</math>: when we add <math>x,y,z</math>, it must be a multiple of <math>9</math> to yield an integer mean. Also remember that the median of an increasing list of <math>9</math> numbers is the fifth number.
  
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~tsun26
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\textbf{Case 1:}
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Since <math>x<1</math>, <math>z<7</math>, <math>x</math> must be the minimum of this list of numbers and <math>7</math> is the max: so the range is <math>7-x</math>. This must be equal to <math>7</math>, so <math>x=0</math>. Now we try to make the median an integer.
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With <math>x=0</math>, the fifth smallest number currently is <math>5.2</math>, which is clearly not an integer. But if we stick <math>y</math> or <math>z</math> in between <math>3.2</math> and <math>5.2</math>, then the median will be an integer. So let one of <math>y,z</math> be <math>4</math>: then the sum of all the numbers we have so far is <math>28.8</math>. To make this a multiple of <math>9</math>, the last number has to be <math>7.2</math> to add up to <math>36</math>. However, we said that <math>z<7</math>, so this violates the bounds of this case. So then we set <math>y=5</math>, so our sum is <math>29.9</math>: then <math>z=36-29.9 = 6.2</math>. Here everything checks out, so for this case there is one solution: <math>(0,5,6.2)</math>.

Revision as of 02:15, 14 November 2024

Problem 10

A list of 9 real numbers consists of $1$, $2.2$, $3.2$, $5.2$, $6.2$, $7$, as well as $x, y,z$ with $x\leq y\leq z$. The range of the list is $7$, and the mean and median are both positive integers. How many ordered triples $(x,y,z)$ are possible?

$\textbf{(A) }1 \qquad\textbf{(B) }2 \qquad\textbf{(C) }3 \qquad\textbf{(D) }4 \qquad\textbf{(E) \text{infinitely many}}\qquad$

Solution

It is easiest to do casework on the range. We have four possible cases: $x<1$, $z<7$ $x>1$, $z<7$ $x<1$, $z>7$ $x>1$, $z>7$

These encapsulate all possible values of $x$ and $z$ we can choose, so we're not leaving anything out. As we'll see, the problem will become simpler this way. Since we want integer mean, first note the sum of the values given to be $24.8$: when we add $x,y,z$, it must be a multiple of $9$ to yield an integer mean. Also remember that the median of an increasing list of $9$ numbers is the fifth number.

\textbf{Case 1:}

Since $x<1$, $z<7$, $x$ must be the minimum of this list of numbers and $7$ is the max: so the range is $7-x$. This must be equal to $7$, so $x=0$. Now we try to make the median an integer. With $x=0$, the fifth smallest number currently is $5.2$, which is clearly not an integer. But if we stick $y$ or $z$ in between $3.2$ and $5.2$, then the median will be an integer. So let one of $y,z$ be $4$: then the sum of all the numbers we have so far is $28.8$. To make this a multiple of $9$, the last number has to be $7.2$ to add up to $36$. However, we said that $z<7$, so this violates the bounds of this case. So then we set $y=5$, so our sum is $29.9$: then $z=36-29.9 = 6.2$. Here everything checks out, so for this case there is one solution: $(0,5,6.2)$.

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