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Difference between revisions of "2018 AMC 10A Problems/Problem 25"

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{{duplicate|[[2018 AMC 12A Problems|2018 AMC 12A #25]] and [[2018 AMC 10A Problems|2018 AMC 10A #25]]}}
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{{duplicate|[[2018 AMC 10A Problems/Problem 25|2018 AMC 10A #25]] and [[2018 AMC 12A Problems/Problem 25|2018 AMC 12A #25]]}}
  
 
== Problem ==
 
== Problem ==
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\left(9c-a^2\right)10^n &= 9b-9c-a^2. &&(\bigstar)
 
\left(9c-a^2\right)10^n &= 9b-9c-a^2. &&(\bigstar)
 
\end{align*}</cmath>
 
\end{align*}</cmath>
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Note that <math>(\bigstar)</math> is a linear equation with <math>10^n.</math> Since it has at least two solutions of <math>n,</math> it has at least two solutions of <math>10^n.</math> We conclude that <math>(\bigstar)</math> must be an identity, so we get the following system of equations:
+
Let <math>y=10^n.</math> Note that <math>(\bigstar)</math> is a linear equation with <math>y,</math> and <math>y</math> is a one-to-one function of <math>n.</math> Since <math>(\bigstar)</math> has at least two solutions of <math>n,</math> it has at least two solutions of <math>y.</math> We conclude that <math>(\bigstar)</math> must be an identity, so we get the following system of equations:
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
 
9c-a^2&=0, \\
 
9c-a^2&=0, \\
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The first equation implies that <math>c=\frac{a^2}{9}.</math> Substituting this into the second equation gives <math>b=\frac{2a^2}{9}.</math>  
 
The first equation implies that <math>c=\frac{a^2}{9}.</math> Substituting this into the second equation gives <math>b=\frac{2a^2}{9}.</math>  
  
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To maximize <math>a + b + c = a + \frac{a^2}{3},</math> we need to maximize <math>a.</math> Clearly, <math>a</math> must be divisible by <math>3.</math> If <math>a=9,</math> then <math>(b,c)=(18,9),</math> which violates the restrictions. However, if <math>a=6,</math> then <math>(b,c)=(8,4).</math> Therefore, the greatest possible value of <math>a + b + c</math> is <math>6+8+4=\boxed{\textbf{(D) } 18}.</math>
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To maximize <math>a + b + c = a + \frac{a^2}{3},</math> we need to maximize <math>a.</math> Clearly, <math>a</math> must be divisible by <math>3.</math> The possibilities for <math>(a,b,c)</math> are <math>(9,18,9),(6,8,4),</math> or <math>(3,2,1),</math> but <math>(9,18,9)</math> is invalid. Therefore, the greatest possible value of <math>a + b + c</math> is <math>6+8+4=\boxed{\textbf{(D) } 18}.</math>
  
 
~CantonMathGuy (Solution)
 
~CantonMathGuy (Solution)
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We now only need to know whether <math>a + b + c = 20</math> might work for any larger <math>n</math>. We will always get equations like <math>100001c - b = 11111a^2</math> where the <math>c</math> coefficient is very close to being nine times the <math>a</math> coefficient. Since the <math>b</math> term will be quite insignificant, we know that once again <math>a^2</math> must equal <math>9c</math>, and thus <math>a = 9, c = 9</math> is our only hope to reach <math>20</math>. Substituting and dividing through by <math>9</math>, we will have something like <math>100001 - \frac{b}{9} = 99999</math>. No matter what <math>n</math> really was, <math>b</math> is out of range (and certainly isn't <math>2</math> as we would have needed).
 
We now only need to know whether <math>a + b + c = 20</math> might work for any larger <math>n</math>. We will always get equations like <math>100001c - b = 11111a^2</math> where the <math>c</math> coefficient is very close to being nine times the <math>a</math> coefficient. Since the <math>b</math> term will be quite insignificant, we know that once again <math>a^2</math> must equal <math>9c</math>, and thus <math>a = 9, c = 9</math> is our only hope to reach <math>20</math>. Substituting and dividing through by <math>9</math>, we will have something like <math>100001 - \frac{b}{9} = 99999</math>. No matter what <math>n</math> really was, <math>b</math> is out of range (and certainly isn't <math>2</math> as we would have needed).
  
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The answer then is <math>\boxed{\textbf{(D)} \text{ 18}}</math>.
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The answer then is <math>\boxed{\textbf{(D) } 18}</math>.
  
 
== Solution 3 ==
 
== Solution 3 ==
  
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The given equation can be written as:
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The given equation can be written as
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<cmath>c \cdot ( \overbrace{1111 \ldots 1111}^\text{2n}) - b \cdot ( \overbrace{11 \ldots 11}^\text{n} ) = a^2 \cdot ( \overbrace{11 \ldots 11}^\text{n} )^2</cmath>
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<cmath>c \cdot (\phantom{ } \overbrace{1111 \ldots 1111}^{2n\text{ digits}}\phantom{ }) - b \cdot (\phantom{ } \overbrace{11 \ldots 11}^{n\text{ digits}} \phantom{ }) = a^2 \cdot (\phantom{ } \overbrace{11 \ldots 11}^{n\text{ digits}} \phantom{ })^2.</cmath>
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Divide by <math>\overbrace{11 \ldots 11}^\text{n}</math> on both sides:
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Divide by <math>\overbrace{11 \ldots 11}^{n\text{ digits}}</math> on both sides:
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<cmath>c \cdot ( \overbrace{1000 \ldots 0001}^\text{n+1}) - b = a^2 \cdot ( \overbrace{11 \ldots 11}^\text{n} )</cmath>
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<cmath>c \cdot (\phantom{ } \overbrace{1000 \ldots 0001}^{n+1\text{ digits}}\phantom{ }) - b = a^2 \cdot (\phantom{ } \overbrace{11 \ldots 11}^{n\text{ digits}} \phantom{ }).</cmath>
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Next, split the first term to make it easier to deal with.
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Next, split the first term to make it easier to deal with:
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<cmath>2c + c \cdot (\overbrace{99 \ldots 99}^\text{n}) - b = a^2 \cdot ( \overbrace{11 \ldots 11}^\text{n} )</cmath>
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<cmath>\begin{align*}
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<cmath>2c - b = (a^2 - 9c) \cdot (\overbrace{11 \ldots 11}^\text{n})</cmath>
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2c + c \cdot (\phantom{ }\overbrace{99 \ldots 99}^{n\text{ digits}}\phantom{ }) - b &= a^2 \cdot (\phantom{ } \overbrace{11 \ldots 11}^{n\text{ digits}} \phantom{ }) \\
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Because <math>2c - b</math> and <math>a^2 - 9c</math> are constants and because there must be at least two distinct values of <math>n</math> that satisfy, <math>2c - b = a^2 - 9c = 0</math>. Thus, we have:
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2c - b &= (a^2 - 9c) \cdot (\phantom{ }\overbrace{11 \ldots 11}^{n\text{ digits}}\phantom{ }).
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<cmath>2c=b</cmath>
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\end{align*}</cmath>
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<cmath>a^2=9c</cmath>
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Because <math>2c - b</math> and <math>a^2 - 9c</math> are constants and because there must be at least two distinct values of <math>n</math> that satisfy, <math>2c - b = a^2 - 9c = 0.</math> Thus, we have
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Knowing that <math>a</math>, <math>b</math>, and <math>c</math> are single digit positive integers and that <math>9c</math> must be a perfect square, the values of <math>(a,b,c)</math> that satisfy both equations are <math>(3,2,1)</math> and <math>(6,8,4).</math> Finally, <math>6 + 8 + 4 = \boxed{\textbf{(D)} \text{18}}</math>.
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<cmath>\begin{align*}
 +
2c&=b, \\
 +
a^2&=9c.
 +
\end{align*}</cmath>
 +
Knowing that <math>a,b,</math> and <math>c</math> are single digit positive integers and that <math>9c</math> must be a perfect square, the values of <math>(a,b,c)</math> that satisfy both equations are <math>(3,2,1)</math> and <math>(6,8,4).</math> Finally, <math>6 + 8 + 4 = \boxed{\textbf{(D) } 18}.</math>
  
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~LegionOfAvatars
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~LegionOfAvatars (Solution)
  
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== Solution 4 (If you are running out of time) ==
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~MRENTHUSIASM (Reformatting)
  
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Considering this is an AMC 10 and calculators are not allowed, the number of digits in <math>A_n</math> and <math>B_n</math>, <math>n</math>, probably is not greater than 3. Checking cases with <math>n=2</math> digits first, we find that if <math>a=9</math> then there are no solutions with a two digit <math>B_n</math>. Thus we check the case with <math>a=8</math> and find that <math>88^2=7744</math>. Thus if <math>c=7</math> and <math>b=3</math> then <math>C_n - B_n = A_n^2</math> for <math>n=2</math>. If <math>n=1</math>, then <math>C_n - B_n = A_n^2</math> if <math>c=7</math>, <math>a=8</math>, and <math>b=3</math>. If this is the case then <math>a+b+c=\boxed{\textbf{(D)} \text{18}}</math>.
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==Solution 4==
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~Dhillonr25
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By [[PaperMath’s sum]], the answer is <math>6+8+4=\boxed{\textbf{(D) } 18}</math>
  
 
== Video Solution by Richard Rusczyk ==
 
== Video Solution by Richard Rusczyk ==

Revision as of 15:50, 21 January 2024

The following problem is from both the 2018 AMC 10A #25 and 2018 AMC 12A #25, so both problems redirect to this page.

Problem

For a positive integer $n$ and nonzero digits $a$, $b$, and $c$, let $A_n$ be the $n$-digit integer each of whose digits is equal to $a$; let $B_n$ be the $n$-digit integer each of whose digits is equal to $b$, and let $C_n$ be the $2n$-digit (not $n$-digit) integer each of whose digits is equal to $c$. What is the greatest possible value of $a + b + c$ for which there are at least two values of $n$ such that $C_n - B_n = A_n^2$?

$\textbf{(A) } 12 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 20$

Solution 1

By geometric series, we have \begin{alignat*}{8} A_n&=a\bigl(\phantom{ }\underbrace{111\cdots1}_{n\text{ digits}}\phantom{ }\bigr)&&=a\left(1+10+10^2+\cdots+10^{n-1}\right)&&=a\cdot\frac{10^n-1}{9}, \\ B_n&=b\bigl(\phantom{ }\underbrace{111\cdots1}_{n\text{ digits}}\phantom{ }\bigr)&&=b\left(1+10+10^2+\cdots+10^{n-1}\right)&&=b\cdot\frac{10^n-1}{9}, \\ C_n&=c\bigl(\phantom{ }\underbrace{111\cdots1}_{2n\text{ digits}}\phantom{ }\bigr)&&=c\left(1+10+10^2+\cdots+10^{2n-1}\right)&&=c\cdot\frac{10^{2n}-1}{9}. \end{alignat*} By substitution, we rewrite the given equation $C_n - B_n = A_n^2$ as \[c\cdot\frac{10^{2n}-1}{9} - b\cdot\frac{10^n-1}{9} = a^2\cdot\left(\frac{10^n-1}{9}\right)^2.\] Since $n > 0,$ it follows that $10^n > 1.$ We divide both sides by $\frac{10^n-1}{9}$ and then rearrange: \begin{align*} c\left(10^n+1\right) - b &= a^2\cdot\frac{10^n-1}{9} \\ 9c\left(10^n+1\right) - 9b &= a^2\left(10^n-1\right) \\ \left(9c-a^2\right)10^n &= 9b-9c-a^2. &&(\bigstar) \end{align*} Let $y=10^n.$ Note that $(\bigstar)$ is a linear equation with $y,$ and $y$ is a one-to-one function of $n.$ Since $(\bigstar)$ has at least two solutions of $n,$ it has at least two solutions of $y.$ We conclude that $(\bigstar)$ must be an identity, so we get the following system of equations: \begin{align*} 9c-a^2&=0, \\ 9b-9c-a^2&=0. \end{align*} The first equation implies that $c=\frac{a^2}{9}.$ Substituting this into the second equation gives $b=\frac{2a^2}{9}.$

To maximize $a + b + c = a + \frac{a^2}{3},$ we need to maximize $a.$ Clearly, $a$ must be divisible by $3.$ The possibilities for $(a,b,c)$ are $(9,18,9),(6,8,4),$ or $(3,2,1),$ but $(9,18,9)$ is invalid. Therefore, the greatest possible value of $a + b + c$ is $6+8+4=\boxed{\textbf{(D) } 18}.$

~CantonMathGuy (Solution)

~MRENTHUSIASM (Revision)

Solution 2

Immediately start trying $n = 1$ and $n = 2$. These give the system of equations $11c - b = a^2$ and $1111c - 11b = (11a)^2$ (which simplifies to $101c - b = 11a^2$). These imply that $a^2 = 9c$, so the possible $(a, c)$ pairs are $(9, 9)$, $(6, 4)$, and $(3, 1)$. The first puts $b$ out of range but the second makes $b = 8$. We now know the answer is at least $6 + 8 + 4 = 18$.

We now only need to know whether $a + b + c = 20$ might work for any larger $n$. We will always get equations like $100001c - b = 11111a^2$ where the $c$ coefficient is very close to being nine times the $a$ coefficient. Since the $b$ term will be quite insignificant, we know that once again $a^2$ must equal $9c$, and thus $a = 9, c = 9$ is our only hope to reach $20$. Substituting and dividing through by $9$, we will have something like $100001 - \frac{b}{9} = 99999$. No matter what $n$ really was, $b$ is out of range (and certainly isn't $2$ as we would have needed).

The answer then is $\boxed{\textbf{(D) } 18}$.

Solution 3

The given equation can be written as \[c \cdot (\phantom{ } \overbrace{1111 \ldots 1111}^{2n\text{ digits}}\phantom{ }) - b \cdot (\phantom{ } \overbrace{11 \ldots 11}^{n\text{ digits}} \phantom{ }) = a^2 \cdot (\phantom{ } \overbrace{11 \ldots 11}^{n\text{ digits}} \phantom{ })^2.\] Divide by $\overbrace{11 \ldots 11}^{n\text{ digits}}$ on both sides: \[c \cdot (\phantom{ } \overbrace{1000 \ldots 0001}^{n+1\text{ digits}}\phantom{ }) - b = a^2 \cdot (\phantom{ } \overbrace{11 \ldots 11}^{n\text{ digits}} \phantom{ }).\] Next, split the first term to make it easier to deal with: \begin{align*} 2c + c \cdot (\phantom{ }\overbrace{99 \ldots 99}^{n\text{ digits}}\phantom{ }) - b &= a^2 \cdot (\phantom{ } \overbrace{11 \ldots 11}^{n\text{ digits}} \phantom{ }) \\ 2c - b &= (a^2 - 9c) \cdot (\phantom{ }\overbrace{11 \ldots 11}^{n\text{ digits}}\phantom{ }). \end{align*} Because $2c - b$ and $a^2 - 9c$ are constants and because there must be at least two distinct values of $n$ that satisfy, $2c - b = a^2 - 9c = 0.$ Thus, we have \begin{align*} 2c&=b, \\ a^2&=9c. \end{align*} Knowing that $a,b,$ and $c$ are single digit positive integers and that $9c$ must be a perfect square, the values of $(a,b,c)$ that satisfy both equations are $(3,2,1)$ and $(6,8,4).$ Finally, $6 + 8 + 4 = \boxed{\textbf{(D) } 18}.$

~LegionOfAvatars (Solution)

~MRENTHUSIASM (Reformatting)

Solution 4

By PaperMath’s sum, the answer is $6+8+4=\boxed{\textbf{(D) } 18}$

Video Solution by Richard Rusczyk

https://artofproblemsolving.com/videos/amc/2018amc10a/470

~ dolphin7

See Also

2018 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 24 		Followed by
Last Problem
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
2018 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 24 		Followed by
Last Problem
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

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