AMC10 2021 A

AMC10 2021 A · Q16

AMC10 2021 A · Q16. It mainly tests Linear equations, Arithmetic sequences basics.

In the following list of numbers, the integer $n$ appears $n$ times in the list for $1 \leq n \leq 200$.\[1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \ldots, 200, 200, \ldots , 200\]What is the median of the numbers in this list?

在下列数字列表中，整数 $n$ 在列表中出现了 $n$ 次，其中 $1 \leq n \leq 200$。 \[1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \ldots, 200, 200, \ldots , 200\] 这个列表中数字的中位数是多少？

(A) 100.5 100.5

(B) 134 134

(D) 150.5 150.5

(E) 167 167

Answer

Correct choice: (C)

正确答案：（C）

Solution

There are $1+2+..+199+200=\frac{(200)(201)}{2}=20100$ numbers in total. Let the median be $k$. We want to find the median $k$ such that \[\frac{k(k+1)}{2}=20100/2,\] or \[k(k+1)=20100.\] Note that $\sqrt{20100} \approx 142$. Plugging this value in as $k$ gives \[\frac{1}{2}(142)(143)=10153.\] $10153-142<10050$, so $142$ is the $152$nd and $153$rd numbers, and hence, our desired answer. $\fbox{(C) 142}$. Note that we can derive $\sqrt{20100} \approx 142$ through the formula \[\sqrt{n} = \sqrt{a+b} \approx \sqrt{a} + \frac{b}{2\sqrt{a} + 1},\] where $a$ is a perfect square less than or equal to $n$. We set $a$ to $19600$, so $\sqrt{a} = 140$, and $b = 500$. We then have $n \approx 140 + \frac{500}{2(140)+1} \approx 142$. ~approximation by ciceronii Note by Fasolinka (use answer choices): Once you know that the answer is in the 140s range (200,000 is around 14^2 times 10^2) by the approximation, it is highly improbable for the answer to be anything but C.

列表中共有 $1+2+\dots+199+200=\frac{(200)(201)}{2}=20100$ 个数字。设中位数为 $k$。我们需要找到满足 \[\frac{k(k+1)}{2}=20100/2,\] 即 \[k(k+1)=20100.\] 注意到 $\sqrt{20100} \approx 142$。将此值代入 $k$ 得 \[\frac{1}{2}(142)(143)=10153.\] $10153-142<10050$，所以 $142$ 是第 $152$ 和第 $153$ 个数字，因此就是我们的答案。\fbox{(C) 142}。注意，我们可以通过公式 \[\sqrt{n} = \sqrt{a+b} \approx \sqrt{a} + \frac{b}{2\sqrt{a} + 1},\] 其中 $a$ 是小于等于 $n$ 的完全平方数，来推导出 $\sqrt{20100} \approx 142$。设 $a=19600$，则 $\sqrt{a}=140$，$b=500$。于是 $n \approx 140 + \frac{500}{2(140)+1} \approx 142$。~approximation by ciceronii Fasolinka 的备注（使用答案选项）：一旦知道答案在 140 左右（200,000 大约是 $14^2$ 乘以 $10^2$），就可以高度确定答案就是 C。

Topics