Skip to main content

2021 AMC 12A Problem 16

Problem 16 of 25IntermediateNumber TheoryProbability & Statistics

In the following list of numbers, the integer nn appears nn times in the list for 1≤n≤200.1 \le n \le 200. 1,1, 2,2, 2,2, 3,3, 3,3, 3,3, 4,4, 4,4, 4,4, 4,4, …,\ldots, 200,200, 200,200, …,\ldots, 200200 What is the median of the numbers in this list?

Answer choices

Show solution

Solution

The list has 1+2+⋯+2001 + 2 + \cdots + 200 =200⋅2012= \dfrac{200\cdot 201}{2} =20100= 20100 terms, so the median is the average of the 1005010050th and 1005110051st terms. The value nn occupies positions up to n(n+1)2.\dfrac{n(n+1)}{2}. Since 141⋅1422=10011\dfrac{141\cdot 142}{2} = 10011 and 142⋅1432=10153,\dfrac{142\cdot 143}{2} = 10153, positions 1001210012 through 1015310153 all equal 142.142. Both middle positions fall in this block, so the median is 142.142. Thus, the correct answer is C.
AoPS wiki

Tagged: triangular number · median (data)

More practice