Question

In the following list of numbers, the integer $n$ appears $n$ times in the list for $1\leq n\leq200.$ $\begin{aligned} &1,2,2,3,3,3,4,4,4,4,\ldots,\\ &200,200,\ldots,200 \end{aligned}$ What is the median of the numbers in this list?

$100.5$

$134$

$142$

$150.5$

$167$

Answer: C

Concepts:median (data)triangular number

Difficulty rating: 1420

Solution:

The list contains $\begin{aligned} 1+2+\cdots+200&=\frac{200\cdot201}{2}\\ &=20100 \end{aligned}$ entries, so its two middle positions are $10050$ and $10051.$

There are $\frac{141\cdot142}{2}=10011$ entries through the last $141,$ and $\frac{142\cdot143}{2}=10153$ entries through the last $142.$ Thus both middle entries are $142,$ so the median is $142.$

Thus, C is the correct answer.

2021 AMC 10A Spring Problem 16

Answer: C

Difficulty rating: 1420

Solution:

Problem 16 in Other Years