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The problem and solutions

I've attempted to solve another AMC 10 problem, and the problem is basically like this:

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 the 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 $s$?

I see that they've used the Gauss sign in solutions, but there are a few parts of it that I don't understand. (I do know that in Gauss sign if [n], then it could be n or larger, but no larger than n+1).

In solution 1, when they first state that [s/2]-19 = [s/5], I don't think it'd work all the time - or would it? And in case 2 (solution 1), they say it's [s/2]=s/2 + 1/2, but shouldn't it rather be s/2+1 ?

And in truth, I don't quite know how to solve this problem, and I do think there are ways to solve it without Gauss signs. Anyone have any ideas?

space
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3 Answers3

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I think they are not using Gauss sign (or floor function). The symbols $\lceil\ \rceil$ refer to the ceiling function; i.e. $\lceil x\rceil = \min\{n\in\mathbb{N}\ |\ x\le n \}$.

AugSB
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You can start by setting up two equations to get an idea of the neighborhood where your solutions appear. $$2C=5D, C-D=19$$

Connor James
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This is quite an interesting question, but it's easy enough to solve without using formulas and complicated notation.

We know that for every 2 jumps that Dash takes, Cozy must make 5 jumps to go the same distance (2 * 5 steps = 5 * 2 steps).

So after 10 steps, Dash has taken 3 jumps less than Cozy.

Applying the same logic,

After 20 steps, Dash takes 6 jumps less than Cozy
After 60 steps, Dash takes 18 jumps less than Cozy

At this point, you can just guess and check to find the answer, since there are only a small amount of numbers to test.

After 61 steps, Dash takes 18 jumps less than Cozy
After 62 steps, Dash takes 18 jumps less than Cozy
After 63 steps, Dash takes 19 jumps less than Cozy
After 64 steps, Dash takes 19 jumps less than Cozy
After 65 steps, Dash takes 20 jumps less than Cozy
After 66 steps, Dash takes 19 jumps less than Cozy
After 67 steps, Dash takes 20 jumps less than Cozy
After 68 steps, Dash takes 20 jumps less than Cozy
After 69 steps, Dash takes 21 jumps less than Cozy
After 70 steps, Dash takes 21 jumps less than Cozy

This can all be computed within minutes, if not less than a minute. After 70 steps, it's apparent that Dash will always take more than 20 jumps less than Cozy, so you might as well stop there.

The three cases where Dash takes 19 less jumps are 63, 64, and 66 steps.

Summing these up yields 193, then 1 + 9 + 3 = 13.

Alex Zhang
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