2022 AMC 12A Problem 5

Below is the professionally curated solution for Problem 5 of the 2022 AMC 12A, 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.

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Concepts:lattice point arithmetic sequence

Difficulty rating: 1350

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 $20?$

$441$

$761$

$841$

$921$

$924$

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 $1+\sum_{k=1}^{20}4k=1+4\cdot\frac{20\cdot21}{2}=1+840=841.$

Thus, the correct answer is C.

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