Difference between revisions of "2004 AIME II Problems/Problem 14"

Latest revision as of 11:57, 18 July 2024

Problem

Consider a string of $n$ $7$ 's, $7777\cdots77,$ into which $+$ signs are inserted to produce an arithmetic expression. For example, $7+77+777+7+7=875$ could be obtained from eight $7$ 's in this way. For how many values of $n$ is it possible to insert $+$ signs so that the resulting expression has value $7000$ ?

Solution 1

Suppose we require $a$ $7$ s, $b$ $77$ s, and $c$ $777$ s to sum up to $7000$ ( $a,b,c \ge 0$ ). Then $7a + 77b + 777c = 7000$ , or dividing by $7$ , $a + 11b + 111c = 1000$ . Then the question is asking for the number of values of $n = a + 2b + 3c$ .

Manipulating our equation, we have $a + 2b + 3c = n = 1000 - 9(b + 12c) \Longrightarrow 0 \le 9(b+12c) < 1000$ . Thus the number of potential values of $n$ is the number of multiples of $9$ from $0$ to $1000$ , or $112$ .

However, we forgot to consider the condition that $a \ge 0$ . For a solution set $(b,c): n=1000-9(b+12c)$ , it is possible that $a = n-2b-3c < 0$ (for example, suppose we counted the solution set $(b,c) = (1,9) \Longrightarrow n = 19$ , but substituting into our original equation we find that $a = -10$ , so it is invalid). In particular, this invalidates the values of $n$ for which their only expressions in terms of $(b,c)$ fall into the inequality $9b + 108c < 1000 < 11b + 111c$ .

For $1000 - n = 9k \le 9(7 \cdot 12 + 11) = 855$ , we can express $k$ in terms of $(b,c): n \equiv b \pmod{12}, 0 \le b \le 11$ and $c = \frac{n-b}{12} \le 7$ (in other words, we take the greatest possible value of $c$ , and then "fill in" the remainder by incrementing $b$ ). Then $11b + 111c \le 855 + 2b + 3c \le 855 + 2(11) + 3(7) = 898 < 1000$ , so these values work.

Similarily, for $855 \le 9k \le 9(8 \cdot 12 + 10) = 954$ , we can let $(b,c) = (k-8 \cdot 12,8)$ , and the inequality $11b + 111c \le 954 + 2b + 3c \le 954 + 2(10) + 3(8) = 998 < 1000$ . However, for $9k \ge 963 \Longrightarrow n \le 37$ , we can no longer apply this approach.

So we now have to examine the numbers on an individual basis. For $9k = 972$ , $(b,c) = (0,9)$ works. For $9k = 963, 981, 990, 999 \Longrightarrow n = 37, 19, 10, 1$ , we find (using that respectively, $b = 11,9,10,11 + 12p$ for integers $p$ ) that their is no way to satisfy the inequality $11b + 111c < 1000$ .

Thus, the answer is $112 - 4 = \boxed{108}$ .

A note: Above, we formulated the solution in a forward manner (the last four paragraphs are devoted to showing that all the solutions we found worked except for the four cases pointed out; in a contest setting, we wouldn't need to be nearly as rigorous). A more natural manner of attacking the problem is to think of the process in reverse, namely seeing that $n \equiv 1 \pmod{9}$ , and noting that small values of $n$ would not work.

Looking at the number $7000$ , we obviously see the maximum number of $7's$ : a string of $1000 \ 7's$ . Then, we see that the minimum is $28 \ 7's: \ 777*9 + 7 = 7000$ . The next step is to see by what interval the value of $n$ increases. Since $777$ is $3 \ 7's, \ 77*10 + 7$ is $21 \ 7's$ , we can convert a $777$ into $77's$ and $7's$ and add $18$ to the value of $n$ . Since we have $9 \ 777's$ to work with, this gives us $28,46,64,82,100,118,136,154,172,190 ( = 28 + 18n | 1\leq n\leq 9)$ as values for $n$ . Since $77$ can be converted into $7*11$ , we can add $9$ to $n$ by converting $77$ into $7's$ . Our $n = 190$ , which has $0 \ 777's \ 90 \ 77's \ 10 7's$ . We therefore can add $9$ to $n \ 90$ times by doing this. All values of $n$ not covered by this can be dealt with with the $n = 46 \ (8 \ 777's \ 10 \ 77's \ 2 \ 7's)$ up to $190$ .

Solution 2

To simplify, replace all the $7$ ’s with $1$ ’s. Because the sum is congruent to $n \pmod 9$ and $1000 \equiv 1 \pmod 9 \implies n \equiv 1 \pmod 9$ Also, $n \leq 1000$ . There are $\big\lfloor \tfrac{1000}{9} \big\rfloor + 1 = 112$ positive integers that satisfy both conditions i.e. $\{1, 10, 19, 28, 37, 46, . . . , 1000\}.$

For $n = 1, 10, 19$ , the greatest sum that is less than or equal to $1000$ is $6 \cdot 111 + 1 = 667 \implies 112-3 = 109.$

Thus $n \geq 28$ and let $S = \{28, 37, 46, . . . , 1000\}$ .

Note that $n=28$ is possible because $9 \cdot 111+1 \cdot 1 = 1000$ .

When $n = 37$ , the greatest sum that is at most $1000$ is $8 \cdot 111+6\cdot 11+1 \cdot 1 = 955$ .

All other elements of $S$ are possible because if any element $n$ of $S$ between $46$ and $991$ is possible, then $(n+ 9)$ must be too.

$\textbf{Case 1:}$ Sum has no $11$ 's

It must have at least one $1$ . If it has exactly one $1$ , there must be nine $111$ ’s and $n = 28$ . Thus, for $n \geq 46$ , the sum has more than one $1$ , so it must have at least $1000 - 8 \cdot 111 = 112$ number of $1$ ’s. For $n \leq 1000$ , at least one $111$ . To show that if $n$ is possible, then $(n + 9)$ is possible, replace a $111$ with $1 + 1 + 1$ , replace eleven $(1 + 1)$ ’s with eleven $11$ ’s, and include nine new $1$ ’s as $+1$ ’s. The sum remains $1000$ .

$\textbf{Case 2:}$ Sum has at least one $11$ .

Replace an $11$ with $1 + 1$ , and include nine new $1$ ’s as $+1$ ’s. Now note that $46$ is possible because $8 \cdot 111 + 10 \cdot 11 + 2 \cdot 1 = 1000$ . Thus all elements of $S$ except $37$ are possible.

Thus there are $\boxed{108}$ possible values for $n$ .

~phoenixfire

Solution 3

It's obvious that we cannot have any number $\ge 7777$ because $7777 > 7000$ so the max number that an occur is $777$

Let's say we have $a$ 777's , $b$ 77's and $c$ 7's

From here we get our required equation as $777a + 77b + 7c = 7000$

Now comes the main problem , one might think that if we find number of $(a,b,c)$ then we're done , but in reality we are over-counting our number of $n$ 's. This is because $n$ is the total number of 7's and from our equation we'll get $n$ as $3a + 2b + c$ (because there are three 7's , two 7's and one 7)

The reason why we're over-counting is because , say $a_1 , b_1 , c_1$ be a solution of our original equation and $a_2 , b_2 , c_2$ be another solution of our original equation , then there can be a possibility that $3a_1 + 2b_1 + c_1 = 3a_2 + 2b_2 + c_2$ where $a_1 \neq a_2 , b_1 \neq b_2 , c_1 \neq c_2$ (example : $2 + 3 + 4 = 1 + 5 + 3$ but $2 \neq 1 , 3 \neq 5 , 4 \neq 3$

We know that $0 \le a \le 9$ , $0 \le b \le 90$ , $0 \le c \le 1000$ The bound on $a$ is easier to handle with , so lets start putting values on $a$ and calculate $b , c , n$ by making cases

Reduced equation : $111a + 11b + c = 1000$

Case 1 : $a = 9$

We get $11b + c = 1 \implies b = 0 , c = 1$ is our only solution thus only $\boxed{1}$ value of $n$

Case 2 : $a = 8$

We get $11b + c = 112 \implies b = 0,1,2,\cdots ,10$ and $c = 112 , 101 , 90,\cdots ,2$ From here we get different $n$ 's as $24 + 112 , 24 + 103 , 24 + 94, \cdots ,24 + 22$ (remember $n = 3a + 2b + c$ and if you have difficulty in understanding how we got $n = 24 + (\cdots)$ then just put the values of $a,b,c$ i am sure you will get it :) )

Let's write the sequences of $n$ 's in a compact form , $T_p = 24 + 22 + 9(p-1)$ (This will be helpful later on) There are $\boxed{11}$ values of $n$

Case 3 : $a = 7$

We get $11b + c = 223 \implies b=0,1,2, \cdots ,20$ and $c = 223,212,201, \cdots ,3$ From here we get different $n$ 's as $21 + 223, 21 + 214 , 21 + 205, \cdots ,21 + 43$

Let's also write the sequences of $n$ 's in a compact form , $T_q = 21 + 43 + 9(q-1)$

Now comes the major part , since we need to find distinct $n$ 's so let's subtract the cases where we find common values , from the total number of values.

To do this we need to make $T_p = T_q \implies p - q = 2$ (after some calculations you'll get $p - q = 2$ ) . Now we know that $0 \le p \le 10$ and $0 \le q \le 20$ so we get $p$ as $10,9,\cdots,2$ and $q$ as $8,7,\cdots,0$ so there are 9 common solutions out of 21(diff values of $q$ ) total , so there are $\boxed{21 - 9}$ values of $n$

Case 4 : $a = 6$

We get $11b + c = 334 \implies b = 0,1,2, \cdots ,30$ and $c = 334,323,312, \cdots ,4$ From here we get the different $n$ 's as $18 + 334,18 + 325,18 + 316, \cdots ,18 + 64$

Compact form of terms is $T_r = 18 + 64 + 9(r-1)$ Now let's repeat the process of eliminating common solutions (one thing to notice is that we've removed common solutions of case 2 from case 3 so if we check case 4 with case 3 we'll remove common solutions from all previous cases and hence we do not have to handle common solutions of case 4 with case 2,1 and in general case X with case X-2,X-3,...2,1 , {basically means checking case X with case X-1 is enough})

$T_q = T_r \implies 21 + 43 + 9(q-1) = 18 + 64 + 9(r-1) \implies q - r = 2$ $\implies q = 20,19,18, \cdots ,2$ and $r = 18,17,16, \cdots ,0$ so there are 19 common solutions out of 31 (diff values of r) so there are $\boxed{31 - 19}$ values of $n$

Now hopefully you've seen a pattern , if not try another case 5 with $a = 5$ you'll get $\boxed{41 - 29}$ values of $n$

Like this we get the value of "distinct" $n$ by summing all the sub-values from the cases that we've solved.

$n = 1 + 11 + (21 - 9) + (31 - 19) + (41 - 29) + \cdots + (91 - 79)$

$\implies n = (1 + 11 + 21 + 31 + 41 + \cdots + 91) - (0 + 0 + 9 + 19 + 29 + \cdots + 79)$

$\implies \boxed{n = 108}$

~MrOreoJuice

@@ Line 1: / Line 1: @@
 == Problem ==
-Consider a string of <math> n </math> <math> 7 </math>'s, <math> 7777\cdots77, </math> into which <math> + </math> signs are inserted to produce an arithmetic expression. For example, <math> 7+77+777+7+7=875 </math> could be obtained from eight <math> 7 </math>'s in this way. For how many values of <math> n </math> is it possible to insert <math> + </math> signs so that the resulting expression has value <math> 7000 </math>?
+Consider a string of <math> n </math> <math> 7 </math>'s, <math> 7777\cdots77, </math> into which <math> + </math> signs are inserted to produce an arithmetic [[expression]]. For example, <math> 7+77+777+7+7=875 </math> could be obtained from eight <math> 7 </math>'s in this way. For how many values of <math> n </math> is it possible to insert <math> + </math> signs so that the resulting expression has value <math> 7000 </math>?
-== Solution ==
+== Solution 1 ==
+Suppose we require <math>a</math> <math>7</math>s, <math>b</math> <math>77</math>s, and <math>c</math> <math>777</math>s to sum up to <math>7000</math> (<math>a,b,c \ge 0</math>). Then <math>7a + 77b + 777c = 7000</math>, or dividing by <math>7</math>, <math>a + 11b + 111c = 1000</math>. Then the question is asking for the number of values of <math>n = a + 2b + 3c</math>.
+Manipulating our equation, we have <math>a + 2b + 3c = n = 1000 - 9(b + 12c) \Longrightarrow 0 \le 9(b+12c) < 1000</math>. Thus the number of potential values of <math>n</math> is the number of multiples of <math>9</math> from <math>0</math> to <math>1000</math>, or <math>112</math>.
+However, we forgot to consider the condition that <math>a \ge 0</math>. For a solution set <math>(b,c): n=1000-9(b+12c)</math>, it is possible that <math>a = n-2b-3c < 0</math> (for example, suppose we counted the solution set <math>(b,c) = (1,9) \Longrightarrow n = 19</math>, but substituting into our original equation we find that <math>a = -10</math>, so it is invalid). In particular, this invalidates the values of <math>n</math> for which their only expressions in terms of <math>(b,c)</math> fall into the inequality <math>9b + 108c < 1000 < 11b + 111c</math>.
+For <math>1000 - n = 9k \le 9(7 \cdot 12 + 11) = 855</math>, we can express <math>k</math> in terms of <math>(b,c): n \equiv b \pmod{12}, 0 \le b \le 11</math> and <math>c = \frac{n-b}{12} \le 7</math> (in other words, we take the greatest possible value of <math>c</math>, and then "fill in" the remainder by incrementing <math>b</math>). Then <math>11b + 111c \le 855 + 2b + 3c \le 855 + 2(11) + 3(7) = 898 < 1000</math>, so these values work.
+Similarily, for <math>855 \le 9k \le 9(8 \cdot 12 + 10) = 954</math>, we can let <math>(b,c) = (k-8 \cdot 12,8)</math>, and the inequality <math>11b + 111c \le 954 + 2b + 3c \le 954 + 2(10) + 3(8) = 998 < 1000</math>. However, for <math>9k \ge 963 \Longrightarrow n \le 37</math>, we can no longer apply this approach.
+So we now have to examine the numbers on an individual basis. For <math>9k = 972</math>, <math>(b,c) = (0,9)</math> works. For <math>9k = 963, 981, 990, 999 \Longrightarrow n = 37, 19, 10, 1</math>, we find (using that respectively, <math>b = 11,9,10,11 + 12p</math> for integers <math>p</math>) that their is no way to satisfy the inequality <math>11b + 111c < 1000</math>.
+Thus, the answer is <math>112 - 4 = \boxed{108}</math>.
+----
+A note: Above, we formulated the solution in a forward manner (the last four paragraphs are devoted to showing that all the solutions we found worked except for the four cases pointed out; in a contest setting, we wouldn't need to be nearly as rigorous). A more natural manner of attacking the problem is to think of the process in reverse, namely seeing that <math>n \equiv 1 \pmod{9}</math>, and noting that small values of <math>n</math> would not work.
+Looking at the number <math>7000</math>, we obviously see the maximum number of <math>7's</math>: a string of <math>1000 \ 7's</math>.  Then, we see that the minimum is <math>28 \ 7's: \ 777*9 + 7 = 7000</math>.  The next step is to see by what interval the value of <math>n</math> increases.  Since <math>777</math> is <math>3 \ 7's, \ 77*10 + 7</math> is <math>21 \ 7's</math>, we can convert a <math>777</math> into <math>77's</math> and <math>7's</math> and add <math>18</math> to the value of <math>n</math>.  Since we have <math>9 \ 777's</math> to work with, this gives us <math>28,46,64,82,100,118,136,154,172,190 ( = 28 + 18n | 1\leq n\leq 9)</math> as values for <math>n</math>.  Since <math>77</math> can be converted into <math>7*11</math>, we can add <math>9</math> to <math>n</math> by converting <math>77</math> into <math>7's</math>.  Our <math>n = 190</math>, which has <math>0 \ 777's \ 90 \ 77's \ 10 7's</math>.  We therefore can add <math>9</math> to <math>n \ 90</math> times by doing this.  All values of <math>n</math> not covered by this can be dealt with with the <math>n = 46 \ (8 \ 777's \ 10 \ 77's \ 2 \ 7's)</math> up to <math>190</math>.
+== Solution 2==
+To simplify, replace all the <math>7</math>’s with <math>1</math>’s.
+Because the sum is congruent to <math>n \pmod 9</math> and <cmath>1000 \equiv 1 \pmod 9 \implies n \equiv 1 \pmod 9</cmath>
+Also, <math>n \leq 1000</math>. There are <math>\big\lfloor \tfrac{1000}{9} \big\rfloor + 1 = 112</math> positive integers that satisfy both conditions i.e. <math>\{1, 10, 19, 28, 37, 46, . . . , 1000\}.</math>
+For <math>n = 1, 10, 19</math>, the greatest sum that is less than or equal to <math>1000</math> is <cmath>6 \cdot 111 + 1 = 667 \implies 112-3 = 109.</cmath>
+Thus <math>n \geq 28</math> and let <math>S = \{28, 37, 46, . . . , 1000\}</math>.
+Note that <math>n=28</math> is possible because <math>9 \cdot 111+1 \cdot 1 = 1000</math>.
+When <math>n = 37</math>, the greatest sum that is at most <math>1000</math> is <math>8 \cdot 111+6\cdot 11+1 \cdot 1 = 955</math>.
+All other elements of <math>S</math> are possible because if any element <math>n</math> of <math>S</math> between <math>46</math> and <math>991</math> is possible, then <math>(n+ 9)</math> must be too.
+<math>\textbf{Case 1:}</math> Sum has no <math>11</math>'s
+It must have at least one <math>1</math>.
+If it has exactly one <math>1</math>, there must be nine <math>111</math>’s and <math>n = 28</math>.
+Thus, for <math>n \geq 46</math>, the sum has more than one <math>1</math>, so it must have at least <math>1000 - 8 \cdot 111 = 112</math> number of <math>1</math>’s.
+For <math>n \leq 1000</math>, at least one <math>111</math>.
+To show that if <math>n</math> is possible, then <math>(n + 9)</math> is possible, replace a <math>111</math> with <math>1 + 1 + 1</math>, replace eleven <math>(1 + 1)</math>’s with eleven <math>11</math>’s, and include nine new <math>1</math>’s as <math>+1</math>’s. The sum remains <math>1000</math>.
+<math>\textbf{Case 2:}</math> Sum has at least one <math>11</math>.
+Replace an <math>11</math> with <math>1 + 1</math>, and include nine new <math>1</math>’s as <math>+1</math>’s.
+Now note that <math>46</math> is possible because <math>8 \cdot 111 + 10 \cdot 11 + 2 \cdot 1 = 1000</math>.
+Thus all elements of <math>S</math> except <math>37</math> are possible.
+Thus there are <math>\boxed{108}</math> possible values for <math>n</math>.
+~phoenixfire
+== Solution 3 ==
+It's obvious that we cannot have any number <math>\ge 7777</math> because <math>7777 > 7000</math> so the max number that an occur is <math>777</math>
+Let's say we have <math>a</math>  777's , <math>b</math>  77's and <math>c</math>  7's
+From here we get our required equation as <math>777a + 77b + 7c = 7000</math>
+Now comes the main problem , one might think that if we find number of <math>(a,b,c)</math> then we're done , but in reality we are over-counting our number of <math>n</math>'s. This is because <math>n</math> is the total number of 7's and from our equation we'll get <math>n</math> as <math>3a + 2b + c</math> (because there are three 7's , two 7's and one 7)
+The reason why we're over-counting is because , say <math>a_1 , b_1 , c_1</math> be a solution of our original equation and <math>a_2 , b_2 , c_2</math> be another solution of our original equation , then there can be a possibility that <math>3a_1 + 2b_1 + c_1 = 3a_2 + 2b_2 + c_2</math> where <math>a_1 \neq a_2 , b_1 \neq b_2 , c_1 \neq c_2</math> (example : <math> 2 + 3 + 4 = 1 + 5 + 3</math> but <math>2 \neq 1 , 3 \neq 5 , 4 \neq 3</math>
+We know that <math>0 \le a \le 9</math> , <math>0 \le b \le 90</math> , <math>0 \le c \le 1000</math>
+The bound on <math>a</math> is easier to handle with , so lets start putting values on <math>a</math> and calculate <math>b , c , n</math> by making cases
+Reduced equation : <math>111a + 11b + c = 1000</math>
+Case 1 : <math>a = 9</math>
+We get <math>11b + c = 1 \implies b = 0 , c = 1</math> is our only solution thus only <math>\boxed{1}</math> value of <math>n</math>
+Case 2 : <math>a = 8</math>
+We get <math>11b + c = 112 \implies b = 0,1,2,\cdots ,10</math> and <math>c = 112 , 101 , 90,\cdots ,2</math> From here we get different <math>n</math>'s as <math>24 + 112 , 24 + 103 , 24 + 94, \cdots ,24 + 22</math>  (remember <math>n = 3a + 2b + c</math> and if you have difficulty in understanding how we got <math>n = 24 + (\cdots)</math> then just put the values of <math>a,b,c</math> i am sure you will get it :) )
+Let's write the sequences of <math>n</math>'s in a compact form , <math>T_p = 24 + 22 + 9(p-1)</math> (This will be helpful later on)
+There are <math>\boxed{11}</math> values of <math>n</math>
+Case 3 : <math>a = 7</math>
+We get <math>11b + c = 223 \implies b=0,1,2, \cdots ,20</math> and <math>c = 223,212,201, \cdots ,3</math> From here we get different <math>n</math>'s as <math>21 + 223, 21 + 214 , 21 + 205, \cdots ,21 + 43</math>
+Let's also write the sequences of <math>n</math>'s in a compact form , <math>T_q = 21 + 43 + 9(q-1)</math>
+Now comes the major part , since we need to find distinct <math>n</math>'s so let's subtract the cases where we find common values , from the total number of values.
+To do this we need to make <math>T_p = T_q \implies p - q = 2</math> (after some calculations you'll get <math>p - q = 2</math>) . Now we know that <math>0 \le p \le 10</math> and <math>0 \le q \le 20</math> so we get <math>p</math> as <math>10,9,\cdots,2</math> and <math>q</math> as <math>8,7,\cdots,0</math> so there are 9 common solutions out of 21(diff values of <math>q</math>) total , so there are <math>\boxed{21 - 9}</math> values of <math>n</math>
+Case 4 : <math>a = 6</math>
+We get <math>11b + c = 334 \implies b = 0,1,2, \cdots ,30</math> and <math>c = 334,323,312, \cdots ,4</math> From here we get the different <math>n</math>'s as <math>18 + 334,18 + 325,18 + 316, \cdots ,18 + 64</math>
+Compact form of terms is <math>T_r = 18 + 64 + 9(r-1)</math>
+Now let's repeat the process of eliminating common solutions (one thing to notice is that we've removed common solutions of case 2 from case 3 so if we check case 4 with case 3 we'll remove common solutions from all previous cases and hence we do not have to handle common solutions of case 4 with case 2,1 and in general case X with case X-2,X-3,...2,1 , {basically means checking case X with case X-1 is enough})
+<math>T_q = T_r \implies 21 + 43 + 9(q-1) = 18 + 64 + 9(r-1) \implies q - r = 2</math>
+<math>\implies q = 20,19,18, \cdots ,2</math> and <math>r = 18,17,16, \cdots ,0</math> so there are 19 common solutions out of 31 (diff values of r) so there are <math>\boxed{31 - 19}</math> values of <math>n</math>
+Now hopefully you've seen a pattern , if not try another case 5 with <math>a = 5</math> you'll get <math>\boxed{41 - 29}</math> values of <math>n</math>
+Like this we get the value of "distinct" <math>n</math> by summing all the sub-values from the cases that we've solved.
+<math>n = 1 + 11 + (21 - 9) + (31 - 19) + (41 - 29) + \cdots + (91 - 79)</math>
+<math>\implies n = (1 + 11 + 21 + 31 + 41 + \cdots + 91) - (0 + 0 + 9 + 19 + 29 + \cdots + 79)</math>
+<math>\implies \boxed{n = 108}</math>
+~MrOreoJuice
 == See also ==
-* [[2004 AIME II Problems]]
+{{AIME box|year=2004|n=II|num-b=13|num-a=15}}
+[[Category:Intermediate Number Theory Problems]]
+{{MAA Notice}}

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