AMC10 2015 B

AMC10 2015 B · Q21

AMC10 2015 B · Q21. It mainly tests Averages (mean), GCD & LCM.

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 $s$?

小猫 Cozy 和小狗 Dash 要爬上一段有若干级台阶的楼梯。不过他们不是一级一级走，而是用跳的。Cozy 每次跳上 2 级（如果需要，他会只跳最后 1 级）。Dash 每次跳上 5 级（如果需要，若剩余台阶少于 5 级，他会把剩下的台阶一次跳完）。已知 Dash 到达楼梯顶部所用的跳跃次数比 Cozy 少 19 次。设 $s$ 为满足条件的所有可能台阶总数之和。问 $s$ 的各位数字之和是多少？

(A) 9 9

(B) 11 11

(D) 13 13

(E) 15 15

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): Assume that there are $t$ steps in this staircase and it took Dash $d+1$ jumps. Then the possible values of $t$ are $5d+1,\,5d+2,\,5d+3,\,5d+4,\,5d+5$. On the other hand, it took Cozy $d+20$ jumps, and $t=2d+39$ or $t=2d+40$. There are 10 possible combinations but only 3 of them lead to integer values of $d$: $t=5d+3=2d+39$, or $t=5d+1=2d+40$, or $t=5d+4=2d+40$. The possible values of $t$ are $63,\,66,$ and $64$, and $s=63+66+64=193$. The answer is $1+9+3=13$.

答案（D）：设这段楼梯共有 $t$ 级台阶，Dash 用了 $d+1$ 次跳跃。那么 $t$ 的可能取值为 $5d+1,\,5d+2,\,5d+3,\,5d+4,\,5d+5$。另一方面，Cozy 用了 $d+20$ 次跳跃，并且 $t=2d+39$ 或 $t=2d+40$。共有 10 种可能的组合，但只有 3 种会使 $d$ 取整数：$t=5d+3=2d+39$，或 $t=5d+1=2d+40$，或 $t=5d+4=2d+40$。因此 $t$ 的可能值为 $63,\,66,\,64$，且 $s=63+66+64=193$。答案为 $1+9+3=13$。

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