Question

星期一，Millie 把一夸脱鸟食倒入喂鸟器，其中 $25\%$ 是小米。此后每天她都再加入一夸脱同样混合比例的鸟食，而且不取出剩下的鸟食。每天鸟只吃掉喂鸟器中小米的 $25\%$ ，但会吃掉所有其他种子。在星期几，Millie 刚放入鸟食后，鸟会发现喂鸟器中超过一半的种子是小米？

On Monday, Millie puts a quart of seeds, $25\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $25\%$ of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet?

星期二

Tuesday

星期三

Wednesday

星期四

Thursday

星期五

Friday

星期六

Saturday

答案：D

知识点：等比数列递推不等式

难度评级：1610

小提示：

每天早上，之前小米的 $\tfrac{3}{4}$ 留下，又加入 $\tfrac{1}{4}$ 夸脱新的小米；其他种子总是 $\tfrac{3}{4}$ 夸脱。

Each morning $\tfrac{3}{4}$ of the previous millet remains and $\tfrac{1}{4}$ quart of new millet is added, while the other seeds always total $\tfrac{3}{4}$ quart

大提示：

$n$ 天后小米量为 $1 - \left(\tfrac{3}{4}\right)^n$ 夸脱；要求它超过 $\tfrac{3}{4}$ 。

After $n$ days the millet is $1 - \left(\tfrac{3}{4}\right)^n$ quart; require this to exceed $\tfrac{3}{4}$

解答：

每天鸟留下小米的 $\dfrac{3}{4}$ ，而 Millie 又加入 $\dfrac{1}{4}$ 夸脱新小米，所以 $n$ 天后小米量为 $\begin{aligned} &\dfrac{1}{4}\left(1 + \dfrac{3}{4} + \cdots + \left(\dfrac{3}{4}\right)^{n-1}\right) \\ &= 1 - \left(\dfrac{3}{4}\right)^n \end{aligned}$ 夸脱。

非小米种子总量始终是 $\dfrac{3}{4}$ 夸脱，所以小米超过一半等价于 $1 - \left(\dfrac{3}{4}\right)^n \gt \dfrac{3}{4}$ ，即 $\left(\dfrac{3}{4}\right)^n \lt \dfrac{1}{4}$ 。

因为 $\left(\dfrac{3}{4}\right)^4 = \dfrac{81}{256} \gt \dfrac{1}{4}$ ，而 $\left(\dfrac{3}{4}\right)^5 = \dfrac{243}{1024} \lt \dfrac{1}{4}$ ，所以首次发生在第 $5$ 天，也就是星期五。

所以正确答案是 D。

Each day the birds leave $\dfrac{3}{4}$ of the millet and Millie adds $\dfrac{1}{4}$ quart of new millet, so after $n$ days the millet is $\begin{aligned} &\dfrac{1}{4}\left(1 + \dfrac{3}{4} + \cdots + \left(\dfrac{3}{4}\right)^{n-1}\right) \\ &= 1 - \left(\dfrac{3}{4}\right)^n \end{aligned}$ quart.

The non-millet seeds always total $\dfrac{3}{4}$ quart, so millet exceeds half when $1 - \left(\dfrac{3}{4}\right)^n \gt \dfrac{3}{4},$ that is, $\left(\dfrac{3}{4}\right)^n \lt \dfrac{1}{4}.$

Since $\left(\dfrac{3}{4}\right)^4 = \dfrac{81}{256} \gt \dfrac{1}{4}$ and $\left(\dfrac{3}{4}\right)^5 = \dfrac{243}{1024} \lt \dfrac{1}{4},$ this first happens on day $5,$ which is Friday.

Thus, the correct answer is D.

2009 AMC 12B 第 11 题

答案：D

难度评级：1610

小提示：

大提示：

解答：

其他年份的第 11 题