Difference between revisions of "2016 AMC 10B Problems/Problem 24"

Latest revision as of 01:50, 5 November 2024

Problem

How many four-digit integers $abcd$ , with $a \neq 0$ , have the property that the three two-digit integers $ab<bc<cd$ form an increasing arithmetic sequence? One such number is $4692$ , where $a=4$ , $b=6$ , $c=9$ , and $d=2$ .

$\textbf{(A)}\ 9\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 20$

Solution

The numbers are $10a+b, 10b+c,$ and $10c+d$ . Note that only $d$ can be zero, the numbers $ab$ , $bc$ , and $cd$ cannot start with a zero, and $a\le b\le c$ .

To form the sequence, we need $(10c+d)-(10b+c)=(10b+c)-(10a+b)$ . This can be rearranged as $10(c-2b+a)=2c-b-d$ . Notice that since the left-hand side is a multiple of $10$ , the right-hand side can only be $0$ or $10$ . (A value of $-10$ would contradict $a\le b\le c$ .) Therefore we have two cases: $a+c-2b=1$ and $a+c-2b=0$ .

Case 1

If $c=9$ , then $b+d=8,\ 2b-a=8$ , so $5\le b\le 8$ . This gives $2593, 4692, 6791, 8890$ . If $c=8$ , then $b+d=6,\ 2b-a=7$ , so $4\le b\le 6$ . This gives $1482, 3581, 5680$ . If $c=7$ , then $b+d=4,\ 2b-a=6$ , so $b=4$ , giving $2470$ . There is no solution for $c=6$ . Added together, this gives us $8$ answers for Case 1.

Case 2

This means that the digits themselves are in an arithmetic sequence, as $a+c-2b=0 \Rightarrow a-b=b-c.$ This gives us $9$ answers, $1234, 1357, 2345, 2468, 3456, 3579, 4567, 5678, 6789.$ Adding the two cases together, we find the answer to be $8+9=$ $\boxed{\textbf{(D) }17}$ .

Solution 2 (Brute Force, when you have lots of time)

Looking at the answer options, all the numbers are pretty small so it is easy to make a list.

$12|23|34 \rightarrow 1234$

$13|35|57 \rightarrow 1357$

$14|48|82 \rightarrow 1482$

$23|34|45 \rightarrow 2345$

$24|46|68 \rightarrow 2468$

$24|47|70 \rightarrow 2470$

$25|59|93 \rightarrow 2593$

$34|45|56 \rightarrow 3456$

$35|57|79 \rightarrow 3579$

$35|58|81 \rightarrow 3581$

$45|56|67 \rightarrow 4567$

$46|69|92 \rightarrow 4692$

$56|67|78 \rightarrow 5678$

$56|68|80 \rightarrow 5680$

$67|78|89 \rightarrow 6789$

$67|79|91 \rightarrow 6791$

$88|89|90 \rightarrow 8890$

Counting all the cases we get our answer of $17$ which is $\boxed{D}$ -srisainandan6

Solution 3

Let $x$ be the difference between the numbers $ab$ , $bc$ , and $cd$ . We then have $10a + b = 10b + c - x$ and $10b + c = 10c + d - x.$

Subtracting the second equation from the first and then simplifying, we are left with: $10a + 8c + d = 19b.$

Notice that $b > c$ . Because the values of $a$ and $d$ are irrelevant compared to the other numbers, we can just find pairs of $(b, c)$ such that $a > 0$ . Trying out each value of $b$ from $2$ to $9$ and summing the number of pairs yields $1 + 2 + 4 + 4 + 3 + 2 + 1 + 0 = \boxed{\textbf{(D) }17}$

- cappucher

Video Solution

https://www.youtube.com/watch?v=UhPxvZ6V4Zs

@@ Line 6: / Line 6: @@
 ==Solution==
-The numbers are <math>10a+b, 10b+c,</math> and <math>10c+d</math>. Note that only <math>d</math> can be zero, and that <math>a\le b\le c</math>.
+The numbers are <math>10a+b, 10b+c,</math> and <math>10c+d</math>. Note that only <math>d</math> can be zero, the numbers <math>ab</math>, <math>bc</math>, and <math>cd</math> cannot start with a zero, and <math>a\le b\le c</math>.
 To form the sequence, we need <math>(10c+d)-(10b+c)=(10b+c)-(10a+b)</math>. This can be rearranged as <math>10(c-2b+a)=2c-b-d</math>. Notice that since the left-hand side is a multiple of <math>10</math>, the right-hand side can only be <math>0</math> or <math>10</math>. (A value of <math>-10</math> would contradict <math>a\le b\le c</math>.) Therefore we have two cases: <math>a+c-2b=1</math> and <math>a+c-2b=0</math>.
-===Case 1===
+'''Case 1'''
 If <math>c=9</math>, then <math>b+d=8,\ 2b-a=8</math>, so <math>5\le b\le 8</math>. This gives <math>2593, 4692, 6791, 8890</math>.
 If <math>c=8</math>, then <math>b+d=6,\ 2b-a=7</math>, so <math>4\le b\le 6</math>. This gives <math>1482, 3581, 5680</math>.
 If <math>c=7</math>, then <math>b+d=4,\ 2b-a=6</math>, so <math>b=4</math>, giving <math>2470</math>.
 There is no solution for <math>c=6</math>.
-===Case 2===
+Added together, this gives us <math>8</math> answers for Case 1.
-This means that the digits themselves are in arithmetic sequence. This gives us <math>9</math> answers, <math>1234, 1357, 2345, 2468, 3456, 3579, 4567, 5678, 6789</math>.
+'''Case 2'''
+This means that the digits themselves are in an arithmetic sequence, as <math>a+c-2b=0 \Rightarrow a-b=b-c.</math> This gives us <math>9</math> answers,
+<cmath>1234, 1357, 2345, 2468, 3456, 3579, 4567, 5678, 6789.</cmath>
+Adding the two cases together, we find the answer to be <math>8+9=</math> <math>\boxed{\textbf{(D) }17}</math>.
+==Solution 2 (Brute Force, when you have lots of time)==
+Looking at the answer options, all the numbers are pretty small so it is easy to make a list.
+<math>12|23|34 \rightarrow 1234</math>
+<math>13|35|57 \rightarrow 1357</math>
+<math>14|48|82 \rightarrow 1482</math>
+<math>23|34|45 \rightarrow 2345</math>
+<math>24|46|68 \rightarrow 2468</math>
+<math>24|47|70 \rightarrow 2470</math>
+<math>25|59|93 \rightarrow 2593</math>
+<math>34|45|56 \rightarrow 3456</math>
+<math>35|57|79 \rightarrow 3579</math>
+<math>35|58|81 \rightarrow 3581</math>
+<math>45|56|67 \rightarrow 4567</math>
+<math>46|69|92 \rightarrow 4692</math>
+<math>56|67|78 \rightarrow 5678</math>
+<math>56|68|80 \rightarrow 5680</math>
+<math>67|78|89 \rightarrow 6789</math>
+<math>67|79|91 \rightarrow 6791</math>
+<math>88|89|90 \rightarrow 8890</math>
+Counting all the cases we get our answer of <math>17</math> which is <math>\boxed{D}</math>
+-srisainandan6
+==Solution 3==
+Let <math>x</math> be the difference between the numbers <math>ab</math>, <math>bc</math>, and <math>cd</math>. We then have
+<cmath>10a + b = 10b + c - x</cmath> and
+<cmath>10b + c = 10c + d - x.</cmath>
+Subtracting the second equation from the first and then simplifying, we are left with:
+<cmath>10a + 8c + d = 19b.</cmath>
+Notice that <math>b > c</math>. Because the values of <math>a</math> and <math>d</math> are irrelevant compared to the other numbers, we can just find pairs of <math>(b, c)</math> such that <math>a > 0</math>. Trying out each value of <math>b</math> from <math>2</math> to <math>9</math> and summing the number of pairs yields <math>1 + 2 + 4 + 4 + 3 + 2 + 1 + 0 = \boxed{\textbf{(D) }17}</math>
+- cappucher
+==Video Solution==
+https://www.youtube.com/watch?v=UhPxvZ6V4Zs
 ==See Also==
 {{AMC10 box|year=2016|ab=B|num-b=23|num-a=25}}
 {{MAA Notice}}

2016 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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