2009 AMC 10B 第 14 题

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?

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：1660

小提示：

每天留下的小米乘以

\dfrac34

，再加入

\dfrac14

夸脱。

Each day the standing millet is multiplied by

\dfrac34,

then

\dfrac14

quart is added

大提示：

其他种子总是

\dfrac34

夸脱；求小米何时首次超过它。

Other seeds always total

\dfrac34

quart; find when millet first exceeds that

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}\text{。}

其他种子总量始终为

\dfrac34

夸脱。小米超过种子总量的一半时，满足

1-\Big(\dfrac34\Big)^n\gt\dfrac34

，即

\Big(\dfrac34\Big)^n\lt\dfrac14

。

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

小提示：

大提示：

解答：

其他年份的第 14 题