Question

星期一，Millie 往喂鸟器中放入一夸脱种子，其中 $25\%$ 是小米。之后每天她都会再加入一夸脱同样配比的种子，不取出剩下的种子。每天鸟只吃掉喂鸟器中小米的 $25\%$ ，但会吃掉所有其他种子。在哪一天，Millie 刚放入种子后，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?

Tuesday

Wednesday

Thursday

Friday

Saturday

答案：D

知识点：等比数列递推不等式

难度评级：1660

解答：

每夸脱新种子加入 $\dfrac14$ 夸脱小米，鸟会留下已有小米的 $\dfrac34$ 。第 $n$ 天加入后的小米量为 $\begin{gathered} \dfrac14\left(1+\dfrac34+\cdots+\Big(\dfrac34\Big)^{n-1}\right) \\ = 1-\Big(\dfrac34\Big)^{n}. \end{gathered}$

它首次超过其他种子总量 $\dfrac34$ 时，满足 $1-\Big(\dfrac34\Big)^n\gt\dfrac34$ ，即 $\Big(\dfrac34\Big)^n\lt\dfrac14$ 。

其他种子总量始终为四分之三夸脱。由于 $\Big(\dfrac34\Big)^4=\dfrac{81}{256}\gt\dfrac14$ ，而 $\Big(\dfrac34\Big)^5=\dfrac{243}{1024}\lt\dfrac14$ ，所以第一次发生在第 $5$ 天，即星期五。

所以正确答案是 D。

Each quart adds $\dfrac14$ quart of millet, and the birds leave $\dfrac34$ of the standing millet. On day $n$ the millet present is $\begin{gathered} \dfrac14\left(1+\dfrac34+\cdots+\Big(\dfrac34\Big)^{n-1}\right) \\ = 1-\Big(\dfrac34\Big)^{n}. \end{gathered}$

The other seeds always total $\dfrac34$ quart. Millet exceeds half when $1-\Big(\dfrac34\Big)^n\gt\dfrac34,$ i.e. $\Big(\dfrac34\Big)^n\lt\dfrac14.$

Since $\Big(\dfrac34\Big)^4=\dfrac{81}{256}\gt\dfrac14$ but $\Big(\dfrac34\Big)^5=\dfrac{243}{1024}\lt\dfrac14,$ this first happens on day $5,$ which is Friday.

Thus, the correct answer is D.

2009 AMC 10B 第 14 题

答案：D

难度评级：1660

解答：

其他年份的第 14 题