2000 AMC 12 Problem 8

Attempt Problem 8 of the 2000 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2000 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

Figures

0,

1,

2,

and

3

consist of

1,

5,

13,

and

25

nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure

100?

10401

19801

20201

39801

40801

Each figure is a diamond whose rows are the odd numbers

1, 3, 5, \ldots

up and back down

Figure

n

has

n^2 + (n + 1)^2

unit squares

Figure

n

is a diamond whose row lengths increase through the odd numbers and back down, giving a total of

n^2 + (n + 1)^2

unit squares. This matches

1, 5, 13, 25

for

n = 0, 1, 2, 3.

Therefore figure

100

has

\begin{aligned} 100^2 + 101^2 &= 10000 + 10201 \\ &= 20201 \end{aligned}

unit squares.

Thus, the correct answer is C.

Problem 8 in other years