2022 AMC 12A Problem 5

Attempt Problem 5 of the 2022 AMC 12A 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 2022 AMC 12A solutions, or check the answer key.

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

Let the taxicab distance between points

(x_1,y_1)

and

(x_2,y_2)

in the coordinate plane be given by

|x_1-x_2|+|y_1-y_2|.

For how many points

P

with integer coordinates is the taxicab distance between

P

and the origin less than or equal to

20?

441

761

841

921

924

Count the lattice points on each “diamond”

|x|+|y|=k

separately

There are

4k

points with

|x|+|y|=k

for

k\ge1,

plus the origin

For each

k\ge1,

the set

|x|+|y|=k

contains exactly

4k

lattice points, and

k=0

gives the single origin.

The total is

\begin{aligned} &1+\sum_{k=1}^{20}4k=1+4\cdot\frac{20\cdot21}{2} \\ &=1+840=841. \end{aligned}

Thus, the correct answer is C.

Problem 5 in other years