Difference between revisions of "2015 AMC 10B Problems/Problem 21"
Mr doudou14 (talk | contribs) (→Solution 3) |
Mr doudou14 (talk | contribs) (→Solution 3) |
||
Line 48: | Line 48: | ||
We know from the problem that Dash goes up 3 steps more than Cozy per jump (assuming they aren't within 4 steps from the top). That means that if Dash takes 19 fewer jumps than Cozy to get to the top of the staircase, the staircase must be at least 57 steps high (3*19=57). We then start using guess-and-check: | We know from the problem that Dash goes up 3 steps more than Cozy per jump (assuming they aren't within 4 steps from the top). That means that if Dash takes 19 fewer jumps than Cozy to get to the top of the staircase, the staircase must be at least 57 steps high (3*19=57). We then start using guess-and-check: | ||
− | <math>\left \lceil {57/2} \right \rceil = 29</math> steps for Cozy | + | <math>\left \lceil {57/2} \right \rceil = 29</math> steps for Cozy, and <math>\left \lceil {57/5} \right \rceil = 12</math> steps for Dash, giving a difference of <math>17 steps</math>. |
− | <math>\left \lceil {57/5} \right \rceil = 12</math> steps for Dash | ||
==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}} |
Revision as of 22:47, 21 December 2017
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 the 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
?
Solutions
Solution 1
We can translate this wordy problem into this simple equation:
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
.
Solution 2
We're looking for natural numbers such that
.
Let's call . We now have
, or
.
Obviously, since , this will not work for any value under 6. 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 3
We know from the problem that Dash goes up 3 steps more than Cozy per jump (assuming they aren't within 4 steps from the top). That means that if Dash takes 19 fewer jumps than Cozy to get to the top of the staircase, the staircase must be at least 57 steps high (3*19=57). We then start using guess-and-check:
steps for Cozy, and
steps for Dash, giving a difference of
.
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.