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 ?
Answer Choices:
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Solution:
Assume that there are steps in this staircase and it took Dash jumps. Then the possible values of are . On the other hand, it took Cozy jumps, and or . There are 10 possible combinations but only 3 of them lead to integer values of , or , or . The possible values of are 63,66 , and 64 , and . The answer is .
The problems on this page are the property of the MAA's American Mathematics Competitions