2019 AMC 10B 第 18 题

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。

|A-B|

是多少？

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

小提示：

研究从一个转向点出发、两步后回到同一方向的映射。

Study the two-step map from one turnaround point back to the same direction

大提示：

在极限点处，应用这个两步映射后位置不变。

At the limiting point, applying that two-step map changes nothing

解答：

设某个向家转向的极限点离家

x

千米。当 Henry 朝健身房走完这一段后，他的位置为

2-\frac{2-x}{4}=\frac32+\frac{x}{4}\text{。}

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}\text{。}

在极限点处，这个两步过程不改变位置，所以

x=\frac38+\frac{x}{16}\text{，}

由此得

x=\frac25

。

另一个极限点就是朝健身房走完后到达的位置：

2-\frac{2-\frac25}{4}=\frac85\text{。}

因此

|A-B|=\frac85-\frac25=\frac65=1\frac15\text{。}

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

小提示：

大提示：

解答：

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