Question

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 $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s?$

$9$

$11$

$12$

$13$

$15$

Answer: D

Concepts:floor and ceiling functions casework

Difficulty rating: 2170

Small Hint:

For $t$ steps, Cozy makes $\left\lceil \tfrac{t}{2} \right\rceil$ jumps and Dash makes $\left\lceil \tfrac{t}{5} \right\rceil$ jumps

Big Hint:

Solve $\left\lceil \tfrac{t}{2} \right\rceil - \left\lceil \tfrac{t}{5} \right\rceil = 19$ and collect every valid $t$

Solution:

A staircase of $t$ steps takes Cozy $\left\lceil \tfrac{t}{2} \right\rceil$ jumps and Dash $\left\lceil \tfrac{t}{5} \right\rceil$ jumps, and we need the difference to equal $19.$

Suppose Dash makes $d+1$ jumps. Then $t$ is one of $5d+1,$ $5d+2,$ $5d+3,$ $5d+4,$ $5d+5.$ Cozy makes $d+20$ jumps, so $t$ is either $2d+39$ or $2d+40.$ Equating these two lists gives an integer $d$ only in the three cases $\begin{gathered} 5d+3=2d+39,\\ 5d+1=2d+40,\\ 5d+4=2d+40. \end{gathered}$ These yield respectively $t=63,66,64.$ Thus the valid values are $t=63,$ $64,$ and $66,$ so $s = 63 + 64 + 66 = 193.$ Its digit sum is $1 + 9 + 3 = 13.$

Thus, the correct answer is D.

2015 AMC 12B Problem 21

Answer: D

Difficulty rating: 2170

Small Hint:

Big Hint:

Solution:

Problem 21 in Other Years