AMC12 2015 B

AMC12 2015 B · Q21

AMC12 2015 B · Q21. It mainly tests GCD & LCM, Diophantine equations (integer solutions).

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 $s$?

猫咪Cozy和狗狗Dash正在爬一个有一定数量台阶的楼梯。然而，他们不是一步一步走台阶，而是跳跃。Cozy每次跳跃上升两级台阶（不过如果必要，他会只跳最后一级）。Dash每次跳跃上升五级台阶（不过如果必要，他会跳剩下的不足5级的台阶）。假设Dash比Cozy少用了19次跳跃到达楼梯顶部。让$s$表示所有可能的台阶数的和。这个楼梯可能有的台阶数的和$s$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

Assume there are $t$ steps and Dash took $d+1$ jumps. Then $t=5d+r$ where $r=1,2,3,4,$ or $5$. Cozy took $d+20$ jumps, so $t=2(d+20)$ or $t=2(d+20)-1=2d+39$ or $2d+40$. The valid cases are $5d+3=2d+39$ ($d=12$, $t=63$), $5d+1=2d+40$ ($d=13$, $t=66$), $5d+4=2d+40$ ($d=12$, $t=64$). Thus $s=193$ and sum of digits is $13$.

假设有$t$级台阶，Dash用了$d+1$次跳跃。那么$t=5d+r$，其中$r=1,2,3,4$或$5$。Cozy用了$d+20$次跳跃，所以$t=2(d+20)$或$t=2(d+20)-1=2d+39$或$2d+40$。有效情况是$5d+3=2d+39$（$d=12$，$t=63$），$5d+1=2d+40$（$d=13$，$t=66$），$5d+4=2d+40$（$d=12$，$t=64$）。因此$s=193$，各位数字和是$13$。

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