Question

猫 Cozy 和狗 Dash 正在爬一段有若干台阶的楼梯。

不过他们不是一步一级，而是跳。

Cozy 每次跳上两级台阶；若最后只剩一级，他会只跳最后一级。Dash 每次跳上 $5$ 级台阶；若最后剩下少于五级，他会直接跳完剩下的台阶。

假设 Dash 到达楼顶所需跳数比 Cozy 少 $19$ 次。令 $s$ 为所有可能台阶数的和。求 $s$ 的各位数字之和。

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 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 $s?$

$9$

$11$

$12$

$13$

$15$

答案：D

知识点：取整函数丢番图方程分类讨论

难度评级：2180

解答：

设 Dash 跳 $d+1$ 次，则台阶数 $t$ 是中的一个。Cozy 多跳 $19$ 次，所以 Cozy 跳 $d+20$ 次，这意味着 $t$ 是 $2d+39$ 或 $2d+40$ 。 $\begin{gathered} 5d+1,5d+2,5d+3, \\ 5d+4,5d+5. \end{gathered}$

匹配这些可能性，整数解来自对应 $t=63,66,64$ 。 $\begin{gathered} 5d+3=2d+39, \\ 5d+1=2d+40, \\ 5d+4=2d+40. \end{gathered}$

因此 $s=63+66+64=193$ ，各位数字之和为 $13$ 。

所以正确答案是 D。

Suppose Dash takes $d+1$ jumps. Then the number of steps $t$ is one of $\begin{gathered} 5d+1,5d+2,5d+3, \\ 5d+4,5d+5. \end{gathered}$ Cozy takes $19$ more jumps, so Cozy takes $d+20$ jumps, which means $t$ is either $2d+39$ or $2d+40$ .

Matching these possibilities, the integer solutions are $\begin{gathered} 5d+3=2d+39, \\ 5d+1=2d+40, \\ 5d+4=2d+40. \end{gathered}$ They give $t=63,66,64$ , respectively.

Thus $s=63+66+64=193$ , and the sum of its digits is $13$ .

Thus, the correct answer is D.

2015 AMC 10B 第 21 题

答案：D

难度评级：2180

解答：

其他年份的第 21 题