2021 AMC 10A Spring Problem 16

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

The list has

1+2+\cdots+200

entries

Locate the two middle positions using triangular numbers

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.

Problem 16 in other years