What is the median of the following list of 4040 numbers? 1, 2, 3, …, 2020, 1^2, 2^2, 3^2, …, 2020^2

2020 AMC 12A Problem 8 — Solution

Below is the professionally curated solution for Problem 8 of the 2020 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2020 AMC 12A solutions, or check the answer key.

Difficulty rating: 1440

What is the median of the following list of $4040$ numbers?

$1, 2, 3, \ldots, 2020, 1^2, 2^2, 3^2, \ldots, 2020^2$

$1974.5$

$1975.5$

$1976.5$

$1977.5$

$1978.5$

The median is the average of the $2020$ th and $2021$ st smallest values.

The perfect squares that are at most $2020$ are $1^2, \ldots, 44^2$ (since $44^2 = 1936$ and $45^2 = 2025$ ), so there are $44$ of them.

Among the list, the numbers $\le 1976$ are the $1976$ integers $1, \ldots, 1976$ together with those $44$ squares, totaling $1976 + 44 = 2020.$

Thus the $2020$ th value is $1976$ and the $2021$ st value is $1977,$ making the median $\dfrac{1976 + 1977}{2} = 1976.5.$

Thus, C is the correct answer.