2015 AMC 12B Problem 21
Problem 21 of 25HarderAlgebraProblem-Solving Techniques
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 that 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
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Solution
A staircase of steps takes Cozy jumps and Dash jumps, and we need the difference to equal
Suppose Dash makes jumps. Then is one of Cozy makes jumps, so is either or Equating these two lists gives an integer only in the three cases These yield respectively Thus the valid values are and so Its digit sum is
Thus, the correct answer is D.
Tagged: floor and ceiling functions · casework