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==Problem== | ==Problem== | ||
− | Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than <math>5</math> steps left). Suppose | + | Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than <math>5</math> steps left). Suppose Dash takes <math>19</math> fewer jumps than Cozy to reach the top of the staircase. Let <math>s</math> denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of <math>s</math>? |
<math>\textbf{(A) }9\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }15</math> | <math>\textbf{(A) }9\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }15</math> | ||
− | ==Solution== | + | ==Solution 1== |
− | 63+64+66 = 193. ( | + | |
+ | Let <math>n</math> be the number of steps. We have | ||
+ | |||
+ | <cmath>\left\lceil \frac{n}{2} \right\rceil - 19 = \left\lceil \frac{n}{5} \right\rceil</cmath> | ||
+ | |||
+ | We will proceed to solve this equation via casework. | ||
+ | |||
+ | Case <math>1</math>: <math>\left\lceil \frac{n}{2} \right\rceil = \frac{n}{2}</math> | ||
+ | |||
+ | Our equation becomes <math>\frac{n}{2} - 19 = \frac{n}{5} + \frac{j}{5}</math>, where <math>j \in \{0,1,2,3,4\}</math> Using the fact that <math>n</math> is an integer, we quickly find that <math>j=1</math> and <math>j=4</math> yield <math>n=64</math> and <math>n=66</math>, respectively. | ||
+ | |||
+ | <br> | ||
+ | |||
+ | Case <math>2</math>: <math>\left\lceil \frac{n}{2} \right\rceil = \frac{n}{2}+\frac{1}{2}</math> | ||
+ | |||
+ | Our equation becomes <math>\frac{n}{2} +\frac{1}{2} - 19 = \frac{n}{5} + \frac{j}{5}</math>, where <math>j \in \{0,1,2,3,4\}</math> Using the fact that <math>n</math> is an integer, we quickly find that <math>j=2</math> yields <math>n=63</math>. Summing up we get <math>63+64+66=193</math>. The sum of the digits is <math>\boxed{\textbf{(D)}\; 13}</math>. | ||
+ | |||
+ | ==Solution 2== | ||
+ | |||
+ | We know from the problem that Dash goes <math>3</math> steps further than Cozy per jump (assuming they aren't within <math>4</math> steps from the top). That means that if Dash takes <math>19</math> fewer jumps than Cozy to get to the top of the staircase, the staircase must be at least <math>3 \cdot 19=57</math> steps high. We then start using guess-and-check: | ||
+ | |||
+ | <math>57</math> steps: <math>\left \lceil {57/2} \right \rceil = 29</math> jumps for Cozy, and <math>\left \lceil {57/5} \right \rceil = 12</math> jumps for Dash, giving a difference of <math>17</math> jumps. | ||
+ | |||
+ | <math>58</math> steps: <math>\left \lceil {58/2} \right \rceil = 29</math> jumps for Cozy, and <math>\left \lceil {58/5} \right \rceil = 12</math> jumps for Dash, giving a difference of <math>17</math> jumps. | ||
+ | |||
+ | <math>59</math> steps: <math>\left \lceil {59/2} \right \rceil = 30</math> jumps for Cozy, and <math>\left \lceil {59/5} \right \rceil = 12</math> jumps for Dash, giving a difference of <math>18</math> jumps. | ||
+ | |||
+ | <math>60</math> steps: <math>\left \lceil {60/2} \right \rceil = 30</math> jumps for Cozy, and <math>\left \lceil {60/5} \right \rceil = 12</math> jumps for Dash, giving a difference of <math>18</math> jumps. | ||
+ | |||
+ | <math>\vdots</math> | ||
+ | |||
+ | By the time we test <math>61</math> steps, we notice that when the number of steps exceeds a multiple of <math>2</math>, the difference in jumps increases. So, we have to find the next number that will increase the difference. <math>62</math> doesn't because both both Cozy's and Dash's number of jumps increases, but <math>63</math> does, and <math>64</math>. <math>65</math> actually gives a difference of <math>20</math> jumps, but <math>66</math> goes back down to <math>19</math> (because Dash had to take another jump when Cozy didn't). We don't need to go any further because the difference will stay above <math>19</math> onward. | ||
+ | |||
+ | Therefore, the possible numbers of steps in the staircase are <math>63</math>, <math>64</math>, and <math>66</math>, giving a sum of <math>193</math>. The sum of those digits is <math>13</math>, so the answer is <math>\boxed{D}</math> | ||
+ | |||
+ | *Minor Edits by Lolgod2 | ||
+ | |||
+ | ==Solution 3== | ||
+ | |||
+ | We're looking for natural numbers <math>x</math> such that <math>\left \lceil{\frac{x}{5}}\right \rceil + 19 = \left \lceil{\frac{x}{2}}\right \rceil</math>. | ||
+ | |||
+ | Let's call <math>x = 10a + b</math>. We now have <math>2a + \left \lceil{\frac{b}{5}}\right \rceil + 19 = 5a + \left \lceil{\frac{b}{2}}\right \rceil</math>, or | ||
+ | |||
+ | <math>19 - 3a = \left \lceil{\frac{b}{2}}\right \rceil - \left \lceil{\frac{b}{5}}\right \rceil</math>. | ||
+ | |||
+ | Obviously, since <math>b \le 10</math>, this will not work for any value of <math>a</math> under <math>6</math>. In addition, since obviously <math>\frac{b}{2} \ge \frac{b}{5}</math>, this will not work for any value over six, so we have <math>a = 6</math> and <math>\left \lceil{\frac{b}{2}}\right \rceil - \left \lceil{\frac{b}{5}}\right \rceil = 1.</math> | ||
+ | |||
+ | This can be achieved when <math>\left \lceil{\frac{b}{5}}\right \rceil = 1</math> and <math>\left \lceil{\frac{b}{2}}\right \rceil = 2</math>, or when <math>\left \lceil{\frac{b}{5}}\right \rceil = 2</math> and <math>\left \lceil{\frac{b}{2}}\right \rceil = 3</math>. | ||
+ | |||
+ | Case One: | ||
+ | |||
+ | We have <math>b \le 5</math> and <math>3 \le b \le 4</math>, so <math>b = 3, 4</math>. | ||
+ | |||
+ | Case Two: | ||
+ | |||
+ | We have <math>6 \le b \le 9</math> and <math>5 \le b \le 6</math>, so <math>b = 6</math>. | ||
+ | |||
+ | We then have <math>63 + 64 + 66 = 193</math>, which has a digit sum of <math>\boxed{13}</math>. | ||
+ | |||
+ | ==Solution 4== | ||
+ | Translate the problem into following equation: | ||
+ | |||
+ | <math>n = 5D - \{0 \sim 4\} = 2C - \{0 \sim 1\}</math> | ||
+ | |||
+ | Since <math>C = D + 19</math>, we have | ||
+ | |||
+ | <math>5D - \{0 \sim 4\} = 2D + 38 - \{0 \sim 1\}</math> | ||
+ | |||
+ | i.e., | ||
+ | |||
+ | <math>3D = 38 + \{0 \sim 4\} - \{0 \sim 1\}</math> | ||
+ | |||
+ | We then have <math>D = 13</math> when <math>\{1\} - \{0\}</math> or <math>\{2\} - \{1\}</math> (the dog's last jump has <math>2</math> steps and the cat's last jump has <math>1</math> step), which yields <math>n = 64</math> and <math>n = 63</math> respectively. | ||
+ | |||
+ | Another solution is <math>D = 14</math> when <math>\{4\} - \{0\}</math>, which yields <math>n = 66</math>. | ||
+ | |||
+ | Therefore, with <math>63 + 64 + 66 = 193</math>, the digit sum is <math>\boxed{13}</math>. | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/TpHZVbBGmVQ | ||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2015|ab=B|num-b=20|num-a=22}} | {{AMC10 box|year=2015|ab=B|num-b=20|num-a=22}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 01:51, 16 June 2022
Problem
Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than steps left). Suppose Dash takes
fewer jumps than Cozy to reach the top of the staircase. Let
denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of
?
Solution 1
Let be the number of steps. We have
We will proceed to solve this equation via casework.
Case :
Our equation becomes , where
Using the fact that
is an integer, we quickly find that
and
yield
and
, respectively.
Case :
Our equation becomes , where
Using the fact that
is an integer, we quickly find that
yields
. Summing up we get
. The sum of the digits is
.
Solution 2
We know from the problem that Dash goes steps further than Cozy per jump (assuming they aren't within
steps from the top). That means that if Dash takes
fewer jumps than Cozy to get to the top of the staircase, the staircase must be at least
steps high. We then start using guess-and-check:
steps:
jumps for Cozy, and
jumps for Dash, giving a difference of
jumps.
steps:
jumps for Cozy, and
jumps for Dash, giving a difference of
jumps.
steps:
jumps for Cozy, and
jumps for Dash, giving a difference of
jumps.
steps:
jumps for Cozy, and
jumps for Dash, giving a difference of
jumps.
By the time we test steps, we notice that when the number of steps exceeds a multiple of
, the difference in jumps increases. So, we have to find the next number that will increase the difference.
doesn't because both both Cozy's and Dash's number of jumps increases, but
does, and
.
actually gives a difference of
jumps, but
goes back down to
(because Dash had to take another jump when Cozy didn't). We don't need to go any further because the difference will stay above
onward.
Therefore, the possible numbers of steps in the staircase are ,
, and
, giving a sum of
. The sum of those digits is
, so the answer is
- Minor Edits by Lolgod2
Solution 3
We're looking for natural numbers such that
.
Let's call . We now have
, or
.
Obviously, since , this will not work for any value of
under
. In addition, since obviously
, this will not work for any value over six, so we have
and
This can be achieved when and
, or when
and
.
Case One:
We have and
, so
.
Case Two:
We have and
, so
.
We then have , which has a digit sum of
.
Solution 4
Translate the problem into following equation:
Since , we have
i.e.,
We then have when
or
(the dog's last jump has
steps and the cat's last jump has
step), which yields
and
respectively.
Another solution is when
, which yields
.
Therefore, with , the digit sum is
.
Video Solution
See Also
2015 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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 |
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