Difference between revisions of "2015 AMC 12B Problems/Problem 21"
Pi over two (talk | contribs) (→Solution) |
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We will proceed to solve this equation via casework. | We will proceed to solve this equation via casework. | ||
| − | Case 1 | + | Case 1: <math>\left\lceil \frac{s}{2} \right\rceil = \frac{s}{2}</math> |
Our equation becomes <math>\frac{s}{2} - 19 = \frac{s}{5} + \frac{j}{5}</math> where <math>j \in \{0,1,2,3,4\}</math> Using the fact that <math>s</math> is an integer, we quickly find that <math>j=1</math> and <math>j=4</math> yield <math>s=64</math> and <math>s=66</math> respectively. | Our equation becomes <math>\frac{s}{2} - 19 = \frac{s}{5} + \frac{j}{5}</math> where <math>j \in \{0,1,2,3,4\}</math> Using the fact that <math>s</math> is an integer, we quickly find that <math>j=1</math> and <math>j=4</math> yield <math>s=64</math> and <math>s=66</math> respectively. | ||
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<br> | <br> | ||
| − | Case 2 | + | Case 2: <math>\left\lceil \frac{s}{2} \right\rceil = \frac{s}{2}+\frac{1}{2}</math> |
Our equation becomes <math>\frac{s}{2} +\frac{1}{2} - 19 = \frac{s}{5} + \frac{j}{5}</math> where <math>j \in \{0,1,2,3,4\}</math> Using the fact that <math>s</math> is an integer, we quickly find that <math>j=2</math> yields <math>s=63</math>. | Our equation becomes <math>\frac{s}{2} +\frac{1}{2} - 19 = \frac{s}{5} + \frac{j}{5}</math> where <math>j \in \{0,1,2,3,4\}</math> Using the fact that <math>s</math> is an integer, we quickly find that <math>j=2</math> yields <math>s=63</math>. | ||
Revision as of 13:13, 5 March 2015
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 5 steps left). Suppose that Dash takes 19 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
We can translate this wordy problem into this simple equation
![\[\left\lceil \frac{s}{2} \right\rceil - 19 = \left\lceil \frac{s}{5} \right\rceil\]](http://latex.artofproblemsolving.com/5/3/9/5399c60725f1698fac5b067a5c7b438e9a88236d.png)
We will proceed to solve this equation via casework.
Case 1: 
Our equation becomes 
where 
Using the fact that is an integer, we quickly find that 
and 
yield 
and 
respectively.
Case 2: 
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 
.
See Also
| 2015 AMC 12B (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 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
