Question

Henry 早上决定去锻炼，他先从家向健身房走了全程的 $\tfrac{3}{4}$ 。健身房离 Henry 家 $2$ 千米。到达该点后，他改变主意，朝家走回当前到家的距离的 $\tfrac{3}{4}$ 。到达新点后，他再次改变主意，朝健身房走当前到健身房距离的 $\tfrac{3}{4}$ 。如果 Henry 一直这样在每次改变主意后朝健身房或家走剩余距离的 $\tfrac{3}{4}$ ，他会越来越接近在离家 $A$ 千米与离家 $B$ 千米的两个点之间来回走。求 $|A-B|$ ？

Henry decides one morning to do a workout, and he walks $\tfrac{3}{4}$ of the way from his home to his gym. The gym is $2$ kilometers away from Henry's home. At that point, he changes his mind and walks $\tfrac{3}{4}$ of the way from where he is back toward home. When he reaches that point, he changes his mind again and walks $\tfrac{3}{4}$ of the distance from there back toward the gym. If Henry keeps changing his mind when he has walked $\tfrac{3}{4}$ of the distance toward either the gym or home from the point where he last changed his mind, he will get very close to walking back and forth between a point $A$ kilometers from home and a point $B$ kilometers from home. What is $|A-B|?$

$\frac{2}{3}$

$1$

$1 \frac{1}{5}$

$1 \frac{1}{4}$

$1 \frac{1}{2}$

答案：C

知识点：递推一次方程

难度评级：1540

解答：

设某次从离家 $x$ 千米的位置向健身房走。此时到健身房的距离为，走完后还离健身房，所以离家距离为 $2-\frac{2-x}{4}=\frac32+\frac{x}{4}.$

接着向家走该距离的四分之三，新位置离家为 $\frac14\left(\frac32+\frac{x}{4}\right)=\frac38+\frac{x}{16}.$

若 Henry 位于其中一个极限点，则经过两段路程后回到同一极限点，所以因此其中一个稳定点离家 $x=\frac25$ 千米。 $x=\frac38+\frac{x}{16},$

用类似思路，从离健身房千米处先向家走、再向健身房走，可得另一个点离家千米。 $2-\frac{2-\frac25}{4}=\frac85.$ $|A-B|=\frac85-\frac25=\frac65=1\frac15.$

因此两点分别离家和千米，差为千米。所以正确答案是 C。

Suppose a limiting homeward turnaround point is $x$ kilometers from home. After Henry walks toward the gym, his position is $2-\frac{2-x}{4}=\frac32+\frac{x}{4}.$

After he turns back toward home, one quarter of that distance remains, so his next homeward turnaround point is $\frac14\left(\frac32+\frac{x}{4}\right)=\frac38+\frac{x}{16}.$

At the limiting point this two-step map leaves the position unchanged, so $x=\frac38+\frac{x}{16},$ which gives $x=\frac25$ .

The other limiting point is the position reached after walking toward the gym: $2-\frac{2-\frac25}{4}=\frac85.$ Therefore $|A-B|=\frac85-\frac25=\frac65=1\frac15.$

Thus, the correct answer is C .

2019 AMC 10B 第 18 题

答案：C

难度评级：1540

解答：

其他年份的第 18 题