Difference between revisions of "2004 AIME I Problems/Problem 15"

Latest revision as of 00:23, 26 December 2022

Problem

For all positive integers $x$ , let $f(x)=\begin{cases}1 & \text{if }x = 1\\ \frac x{10} & \text{if }x\text{ is divisible by 10}\\ x+1 & \text{otherwise}\end{cases}$ and define a sequence as follows: $x_1=x$ and $x_{n+1}=f(x_n)$ for all positive integers $n$ . Let $d(x)$ be the smallest $n$ such that $x_n=1$ . (For example, $d(100)=3$ and $d(87)=7$ .) Let $m$ be the number of positive integers $x$ such that $d(x)=20$ . Find the sum of the distinct prime factors of $m$ .

Solution

Solution 1

We backcount the number of ways. Namely, we start at $x_{20} = 1$ , which can only be reached if $x_{19} = 10$ , and then we perform $18$ operations that either consist of $A: (-1)$ or $B: (\times 10)$ . We represent these operations in a string format, starting with the operation that sends $f(x_{18}) = x_{19}$ and so forth downwards. There are $2^9$ ways to pick the first $9$ operations; however, not all $9$ of them may be $A: (-1)$ otherwise we return back to $x_{10} = 1$ , contradicting our assumption that $20$ was the smallest value of $n$ . Using complementary counting, we see that there are only $2^9 - 1$ ways.

Since we performed the operation $B: (\times 10)$ at least once in the first $9$ operations, it follows that $x_{10} \ge 20$ , so that we no longer have to worry about reaching $1$ again.

However, we must also account for a sequence of $10$ or more $A: (-1)$ s in a row, because that implies that at least one of those numbers was divisible by $10$ , yet the $\frac{x}{10}$ was never used, contradiction. We must use complementary counting again by determining the number of strings of $A,B$ s of length $18$ such that there are $10$ $A$ s in a row. The first ten are not included since we already accounted for that scenario above, so our string of $10$ $A$ s must be preceded by a $B$ . There are no other restrictions on the remaining seven characters. Letting $\square$ to denote either of the functions, and $^{[k]}$ to indicate that the character appears $k$ times in a row, our bad strings can take the forms: $\begin{align*} &\underbrace{BA^{[10]}}\square \square \square \square \square \square \square\\ &\square \underbrace{BA^{[10]}}\square \square \square \square \square \square \\ &\qquad \quad \cdots \quad \cdots \\ &\square \square \square \square \square \square \underbrace{BA^{[10]}} \square \\ &\square \square \square \square \square \square \square \underbrace{BA^{[10]}} \\ \end{align*}$

There are $2^7$ ways to select the operations for the $7$ $\square$ s, and $8$ places to place our $BA^{[10]}$ block. Thus, our answer is $2^9(2^9-1)-8\cdot 2^7 = 2^{18} - 2^9 - 8 \cdot 2^7 = 2^9 \times 509$ , and the answer is $\boxed{511}$ .

Note: This solution is quick and most similar to the official solution; however, neither this nor the official solution prove that the final results of these inverted operations are all distinct. A more sophisticated argument, such as the one below, is needed to do so.

Solution 2

We approach the problem by recursion. We partition the positive integers into the sets $\mathcal{A}_{n,k}=\{x\in\mathbb{Z}^+\,:\, d(x)=n\text{ and } x\equiv k\pmod{10}\}.$ First, we note that $\mathcal{A}_{1,1}=\{1\}$ , so by the disjointness of the $\mathcal{A}_{n,k}$ 's, we know that $1$ is not in any of the other sets. Also, we note that $\mathcal{A}_{1,k}=\emptyset$ for $k=0,2,3,4,5,6,7,8,9$ .

We claim that if $2\le k\le 9$ and $n\ge2$ , then $|\mathcal{A}_{n,k}|=|\mathcal{A}_{n-1,k+1}|$ . To prove this, we show that $f$ (the given function) maps $\mathcal{A}_{n,k}$ bijectively onto $\mathcal{A}_{n-1,k+1}$ . If $x\in \mathcal{A}_{n,k}$ , then $x\equiv k\pmod{10}$ , and $x\ne 1$ , so $f(x)=x+1\equiv k+1\pmod{10}$ . Also, $f^{(n)}(x)=1$ , where $n$ is the smallest positive integer for which this is true. Therefore, $f^{{n-1}}(f(x))=1$ , where $n-1$ is the smallest integer for which this is true. Thus $f(x)\in\mathcal{A}_{n-1,k+1}$ . Also, since $f(x)=x+1$ on this set, we see that $f(x)=f(y)$ implies that $x=y$ . Hence $f$ is an injection. If $y\in\mathcal{A}_{n-1,k+1}$ , then $y-1\equiv k\pmod{10}$ , where $2\le k\le 9$ , so we know that $f(y-1)=y$ , and $y-1\in \mathcal{A}_{n,k}$ . Therefore, $f$ is a surjection, so it must be a bijection. Therefore, if $2\le k\le 9$ and $n\ge2$ , then $|\mathcal{A}_{n,k}|=|\mathcal{A}_{n-1,k+1}|$ .

We also claim that if $k=1$ , $n\ge 2$ and $n\ne 11$ , then $|\mathcal{A}_{n,k}|=|\mathcal{A}_{n-1,k+1}|$ . The proof is the same as in the previous paragraph, though some additional clarification is needed to prove that $f$ is a surjection. If $y\in\mathcal{A}_{n-1,k+1}$ , or rather $y-1\equiv k\equiv 1\pmod{10}$ , then if $y=2$ , we see that $y-1=1$ , and then $f(y-1)=1$ , not $y$ as in the prior argument. However, this does not happen if $n\ne 11$ . It is easy to check that $2\in\mathcal{A}_{10,2}$ . Therefore, the only time that the above argument could fail is if $n-1=10$ and $k+1=2$ . But in every other case, $|\mathcal{A}_{n,k}|=|\mathcal{A}_{n-1,k+1}|$ .

Next, we claim that if $n\ge 3$ , then $|\mathcal{A}_{n,0}|=\Big|\bigcup_{k=0}^9\mathcal{A}_{n-1,k}\Big|$ If $x\in\mathcal{A}_{n,0}$ , then $f(x)=\tfrac{x}{10}$ , which is clearly an injective map. Also, $f^{(n)}(x)=1$ , where $n$ is the smallest positive integer for which this is true. Therefore, $f^{{n-1}}(f(x))=1$ , where $n-1$ is the smallest integer for which this is true. Thus $f(x)\in\mathcal{A}_{n-1,k}$ for some $k$ . Conversely, if $y\in \mathcal{A}_{n-1,k}$ , then $f(10y)=y$ , so $d(10y)=n$ , so $10y\in\mathcal{A}_{n,0}$ .

Based on these bijections, if we let $A_{n,k}=|\mathcal{A}_{n,k}|$ , then $\begin{align*} A_{n,0}&=\sum_{k=0}^{9} A_{n-1,k}\\ A_{n,1}&=A_{n-1,2}\qquad\text{if }n\ne 11\\ A_{n,2}&=A_{n-1,3}\\ A_{n,3}&=A_{n-1,4}\\ &\vdots\\ A_{n,8}&=A_{n-1,9}\\ A_{n,9}&=A_{n-1,0}. \end{align*}$ Let $S_n=\sum_{k=0}^9 A_{n,k}$ . Then by adding the above equations (valid if $n\ne 11$ ), we find that $S_n=2S_{n-1}-A_{n-1,1}.$ Now by using the above relations repeatedly, we find $A_{n-1,1}=A_{n-2,2}=A_{n-3,3}=\cdots=A_{n-9,9}=A_{n-10,0}=S_{n-11}.$ The first relation will only be valid if $n\ne 12$ . Therefore, $S_n=2S_{n-1}-S_{n-11}$ for $n\ge 13$ . For smaller values, it is easy to use the relations to compute that the terms are powers of $2$ , although we note that $A_{11,1}=0$ because of the failure of the above argument. $\begin{array}{c|cccccccccc|c} n&A_{n,1}&A_{n,2}&A_{n,3}&A_{n,4}&A_{n,5}&A_{n,6}&A_{n,7}&A_{n,8}&A_{n,9}&A_{n,0}&S_{n}\\\hline 1&1&0&0&0&0&0&0&0&0&0&1\\ 2&0&0&0&0&0&0&0&0&0&1&1\\ 3&0&0&0&0&0&0&0&0&1&1&2\\ 4&0&0&0&0&0&0&0&1&1&2&4\\ 5&0&0&0&0&0&0&1&1&2&4&8\\ 6&0&0&0&0&0&1&1&2&4&8&16\\ 7&0&0&0&0&1&1&2&4&8&16&32\\ 8&0&0&0&1&1&2&4&8&16&32&64\\ 9&0&0&1&1&2&4&8&16&32&64&128\\ 10&0&1&1&2&4&8&16&32&64&128&256\\ 11&\boxed{0}&1&2&4&8&16&32&64&128&256&511\\ 12&1&2&4&8&16&32&64&128&256&511&1022\\ \end{array}$ After this, we can simply use the recurrence relation for $S_n$ , finding $\begin{array}{c|l} n&S_n\\\hline 1&1\\ 2&1\\ 3&2^1\\ 4&2^2\\ 5&2^3\\ 6&2^4\\ 7&2^5\\ 8&2^6\\ 9&2^7\\ 10&2^8\\ 11&2^9-1\\ 12&2^{10}-2\\ 13&2^{11}-1\cdot 5\\ 14&2^{12}-2\cdot 6\\ 15&2^{13}-4\cdot 7\\ 16&2^{14}-8\cdot 8\\ 17&2^{15}-16\cdot 9\\ 18&2^{16}-32\cdot 10\\ 19&2^{17}-64\cdot 11\\ 20&2^{18}-128\cdot 12. \end{array}$ Therefore, there are $2^{18}- 2^9\cdot 3$ positive integers $x$ with $d(x)=20$ . This factors as $2^9(2^{9}-3)=2^9(509)$ , where $509$ is prime. Thus the answer is $\boxed{511}$ .

@@ Line 2: / Line 2: @@
 For all positive integers <math>x</math>, let
 <cmath>
-f(x)=\begin{cases}1 & \text{if x = 1}}\\ \frac x{10} & \text{if x is divisible by 10}\\ x+1 & \text{otherwise}\end{cases}
+f(x)=\begin{cases}1 & \text{if }x = 1\\ \frac x{10} & \text{if }x\text{ is divisible by 10}\\ x+1 & \text{otherwise}\end{cases}
 </cmath>
 and define a [[sequence]] as follows: <math>x_1=x</math> and <math>x_{n+1}=f(x_n)</math> for all positive integers <math>n</math>. Let <math>d(x)</math> be the smallest <math>n</math> such that <math>x_n=1</math>. (For example, <math>d(100)=3</math> and <math>d(87)=7</math>.) Let <math>m</math> be the number of positive integers <math>x</math> such that <math>d(x)=20</math>. Find the sum of the distinct prime factors of <math>m</math>.
 == Solution ==
+=== Solution 1 ===
 We backcount the number of ways. Namely, we start at <math>x_{20} = 1</math>, which can only be reached if <math>x_{19} = 10</math>, and then we perform <math>18</math> operations that either consist of <math>A: (-1)</math> or <math>B: (\times 10)</math>. We represent these operations in a string format, starting with the operation that sends <math>f(x_{18}) = x_{19}</math> and so forth downwards. There are <math>2^9</math> ways to pick the first <math>9</math> operations; however, not all <math>9</math> of them may be <math>A: (-1)</math> otherwise we return back to <math>x_{10} = 1</math>, contradicting our assumption that <math>20</math> was the smallest value of <math>n</math>. Using [[complementary counting]], we see that there are only <math>2^9 - 1</math> ways.
-Since we performed the operation <math>B: (\times 10)</math> at least once in the first <math>9</math> operations, it follows that <math>x_{10} \ge 20</math>, so that we no longer have to worry about reaching <math>1</math> again. Thus the remaining <math>9</math> operations can be picked in <math>2^9</math> ways, with a total of <math>2^9(2^9 - 1) = 2^{18} - 2^9</math> strings.
+Since we performed the operation <math>B: (\times 10)</math> at least once in the first <math>9</math> operations, it follows that <math>x_{10} \ge 20</math>, so that we no longer have to worry about reaching <math>1</math> again.
-However, we must also account for a sequence of <math>10</math> or more <math>A: (-1)</math>s in a row, because that implies that at least one of those numbers was divisible by <math>10</math>, yet the <math>\frac{x}{10}</math> was never used, contradiction. We must use complement counting again by determining the number of strings of <math>A,B</math>s of length <math>18</math> such that there are <math>10</math> <math>A</math>s in a row. The first ten are not included since we already accounted for that scenario above, so our string of <math>10</math> <math>A</math>s must be preceded by a <math>B</math>. There are no other restrictions on the remaining seven characters. Letting <math>\square</math> to denote either of the functions, and <math>^{[k]}</math> to indicate that the character appears <math>k</math> times in a row, then our bad strings can take the forms:
+However, we must also account for a sequence of <math>10</math> or more <math>A: (-1)</math>s in a row, because that implies that at least one of those numbers was divisible by <math>10</math>, yet the <math>\frac{x}{10}</math> was never used, contradiction. We must use complementary counting again by determining the number of strings of <math>A,B</math>s of length <math>18</math> such that there are <math>10</math> <math>A</math>s in a row. The first ten are not included since we already accounted for that scenario above, so our string of <math>10</math> <math>A</math>s must be preceded by a <math>B</math>. There are no other restrictions on the remaining seven characters. Letting <math>\square</math> to denote either of the functions, and <math>^{[k]}</math> to indicate that the character appears <math>k</math> times in a row, our bad strings can take the forms:
-<center><math>\begin{align*}
+<cmath>\begin{align*}
-&\underbrace{BA^{[10]}}\square \square \square \square \square \square \square \square \\
+&\underbrace{BA^{[10]}}\square \square \square \square \square \square \square\\
-&\square \underbrace{BA^{[10]}}\square \square \square \square \square \square \square \\
+&\square \underbrace{BA^{[10]}}\square \square \square \square \square \square \\
 &\qquad \quad \cdots \quad \cdots \\
-&\square \square \square \square \square \square \square \underbrace{BA^{[10]}} \square \\
+&\square \square \square \square \square \square \underbrace{BA^{[10]}} \square \\
-&\square \square \square \square \square \square \square \square \underbrace{BA^{[10]}} \\
+&\square \square \square \square \square \square \square \underbrace{BA^{[10]}} \\
-\end{align*}</math></center>
+\end{align*}</cmath>
-There are <math>2^7</math> ways to select the operations for the <math>7</math> <math>\square</math>s, and <math>8</math> places to place our <math>BA^{[10]}</math> block. Thus, our answer is <math>2^{18} - 2^9 - 8 \cdot 2^7 = 2^9 \times 509</math>, and the answer is <math>\boxed{511}</math>.
+There are <math>2^7</math> ways to select the operations for the <math>7</math> <math>\square</math>s, and <math>8</math> places to place our <math>BA^{[10]}</math> block. Thus, our answer is <math>2^9(2^9-1)-8\cdot 2^7 = 2^{18} - 2^9 - 8 \cdot 2^7 = 2^9 \times 509</math>, and the answer is <math>\boxed{511}</math>.
+'''Note:''' This solution is quick and most similar to the official solution; however, neither this nor the official solution prove that the final results of these inverted operations are all distinct. A more sophisticated argument, such as the one below, is needed to do so.
+=== Solution 2 ===
+We approach the problem by [[recursion]]. We [[partition]] the positive integers into the sets
+<cmath>\mathcal{A}_{n,k}=\{x\in\mathbb{Z}^+\,:\, d(x)=n\text{ and } x\equiv k\pmod{10}\}.</cmath>
+First, we note that <math>\mathcal{A}_{1,1}=\{1\}</math>, so by the [[disjoint]]ness of the <math>\mathcal{A}_{n,k}</math>'s, we know that <math>1</math> is not in any of the other sets. Also, we note that <math>\mathcal{A}_{1,k}=\emptyset</math> for <math>k=0,2,3,4,5,6,7,8,9</math>.
+We claim that if <math>2\le k\le 9</math> and <math>n\ge2</math>, then <math>|\mathcal{A}_{n,k}|=|\mathcal{A}_{n-1,k+1}|</math>. To prove this, we show that <math>f</math> (the given function) maps <math>\mathcal{A}_{n,k}</math> bijectively onto <math>\mathcal{A}_{n-1,k+1}</math>. If <math>x\in \mathcal{A}_{n,k}</math>, then <math>x\equiv k\pmod{10}</math>, and <math>x\ne 1</math>, so <math>f(x)=x+1\equiv k+1\pmod{10}</math>. Also, <math>f^{(n)}(x)=1</math>, where <math>n</math> is the smallest positive integer for which this is true. Therefore, <math>f^{{n-1}}(f(x))=1</math>, where <math>n-1</math> is the smallest integer for which this is true. Thus <math>f(x)\in\mathcal{A}_{n-1,k+1}</math>. Also, since <math>f(x)=x+1</math> on this set, we see that <math>f(x)=f(y)</math> implies that <math>x=y</math>. Hence <math>f</math> is an [[injection]]. If <math>y\in\mathcal{A}_{n-1,k+1}</math>, then <math>y-1\equiv k\pmod{10}</math>, where <math>2\le k\le 9</math>, so we know that <math>f(y-1)=y</math>, and <math>y-1\in \mathcal{A}_{n,k}</math>. Therefore, <math>f</math> is a [[surjection]], so it must be a [[bijection]]. Therefore,  if <math>2\le k\le 9</math> and <math>n\ge2</math>, then <math>|\mathcal{A}_{n,k}|=|\mathcal{A}_{n-1,k+1}|</math>.
+We also claim that if <math>k=1</math>, <math>n\ge 2</math> and <math>n\ne 11</math>, then  <math>|\mathcal{A}_{n,k}|=|\mathcal{A}_{n-1,k+1}|</math>. The proof is the same as in the previous paragraph, though some additional clarification is needed to prove that <math>f</math> is a surjection. If <math>y\in\mathcal{A}_{n-1,k+1}</math>, or rather <math>y-1\equiv k\equiv 1\pmod{10}</math>, then if <math>y=2</math>, we see that <math>y-1=1</math>, and then <math>f(y-1)=1</math>, not <math>y</math> as in the prior argument. However, this does not happen if <math>n\ne 11</math>. It is easy to check that <math>2\in\mathcal{A}_{10,2}</math>. Therefore, the only time that the above argument could fail is if <math>n-1=10</math> and <math>k+1=2</math>. But in every other case, <math>|\mathcal{A}_{n,k}|=|\mathcal{A}_{n-1,k+1}|</math>.
+Next, we claim that if <math>n\ge 3</math>, then
+<cmath>|\mathcal{A}_{n,0}|=\Big|\bigcup_{k=0}^9\mathcal{A}_{n-1,k}\Big|</cmath>
+If <math>x\in\mathcal{A}_{n,0}</math>, then <math>f(x)=\tfrac{x}{10}</math>, which is clearly an injective map.  Also, <math>f^{(n)}(x)=1</math>, where <math>n</math> is the smallest positive integer for which this is true. Therefore, <math>f^{{n-1}}(f(x))=1</math>, where <math>n-1</math> is the smallest integer for which this is true. Thus <math>f(x)\in\mathcal{A}_{n-1,k}</math> for some <math>k</math>. Conversely, if <math>y\in \mathcal{A}_{n-1,k}</math>, then <math>f(10y)=y</math>, so <math>d(10y)=n</math>, so <math>10y\in\mathcal{A}_{n,0}</math>.
+Based on these bijections, if we let <math>A_{n,k}=|\mathcal{A}_{n,k}|</math>, then
+<cmath>\begin{align*}
+A_{n,0}&=\sum_{k=0}^{9} A_{n-1,k}\\
+A_{n,1}&=A_{n-1,2}\qquad\text{if }n\ne 11\\
+A_{n,2}&=A_{n-1,3}\\
+A_{n,3}&=A_{n-1,4}\\
+&\vdots\\
+A_{n,8}&=A_{n-1,9}\\
+A_{n,9}&=A_{n-1,0}.
+\end{align*}</cmath>
+Let <math>S_n=\sum_{k=0}^9 A_{n,k}</math>. Then by adding the above equations (valid if <math>n\ne 11</math>), we find that
+<cmath>S_n=2S_{n-1}-A_{n-1,1}.</cmath>
+Now by using the above relations repeatedly, we find
+<cmath>A_{n-1,1}=A_{n-2,2}=A_{n-3,3}=\cdots=A_{n-9,9}=A_{n-10,0}=S_{n-11}.</cmath>
+The first relation will only be valid if <math>n\ne 12</math>. Therefore, <math>S_n=2S_{n-1}-S_{n-11}</math> for <math>n\ge 13</math>. For smaller values, it is easy to use the relations to compute that the terms are powers of <math>2</math>, although we note that <math>A_{11,1}=0</math> because of the failure of the above argument.
+<cmath>\begin{array}{c|cccccccccc|c}
+n&A_{n,1}&A_{n,2}&A_{n,3}&A_{n,4}&A_{n,5}&A_{n,6}&A_{n,7}&A_{n,8}&A_{n,9}&A_{n,0}&S_{n}\\\hline
+&1&0&0&0&0&0&0&0&0&0&1\\
+&0&0&0&0&0&0&0&0&0&1&1\\
+&0&0&0&0&0&0&0&0&1&1&2\\
+&0&0&0&0&0&0&0&1&1&2&4\\
+&0&0&0&0&0&0&1&1&2&4&8\\
+&0&0&0&0&0&1&1&2&4&8&16\\
+&0&0&0&0&1&1&2&4&8&16&32\\
+&0&0&0&1&1&2&4&8&16&32&64\\
+&0&0&1&1&2&4&8&16&32&64&128\\
+&0&1&1&2&4&8&16&32&64&128&256\\
+&\boxed{0}&1&2&4&8&16&32&64&128&256&511\\
+&1&2&4&8&16&32&64&128&256&511&1022\\
+\end{array}</cmath>
+After this, we can simply use the recurrence relation for <math>S_n</math>, finding
+<cmath>\begin{array}{c|l}
+n&S_n\\\hline
+&1\\
+&1\\
+&2^1\\
+&2^2\\
+&2^3\\
+&2^4\\
+&2^5\\
+&2^6\\
+&2^7\\
+&2^8\\
+&2^9-1\\
+&2^{10}-2\\
+&2^{11}-1\cdot 5\\
+&2^{12}-2\cdot 6\\
+&2^{13}-4\cdot 7\\
+&2^{14}-8\cdot 8\\
+&2^{15}-16\cdot 9\\
+&2^{16}-32\cdot 10\\
+&2^{17}-64\cdot 11\\
+&2^{18}-128\cdot 12.
+\end{array}</cmath>
+Therefore, there are <math>2^{18}- 2^9\cdot 3</math> positive integers <math>x</math> with <math>d(x)=20</math>. This factors as <math>2^9(2^{9}-3)=2^9(509)</math>, where <math>509</math> is prime. Thus the answer is <math>\boxed{511}</math>.
 == See also ==
@@ Line 27: / Line 99: @@
 [[Category:Intermediate Combinatorics Problems]]
 [[Category:Intermediate Number Theory Problems]]
+{{MAA Notice}}

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