2022 AMC 12A Problem 5

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?

Show solution

Solution

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.