AMC12 2022 A

AMC12 2022 A · Q5

AMC12 2022 A · Q5. It mainly tests Basic counting (rules of product/sum), Coordinate geometry.

The taxicab distance between points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane is 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 $20$?

坐标平面中点$(x_1, y_1)$与$(x_2, y_2)$之间的出租车距离为 \[|x_1 - x_2| + |y_1 - y_2|.\] 有多少个具有整数坐标的点$P$，使得$P$与原点的出租车距离小于或等于$20$？

(A) \, 441 \, 441

(B) \, 761 \, 761

(D) \, 921 \, 921

(E) \, 924 \, 924

Answer

Correct choice: (C)

正确答案：（C）

Solution

Let us consider the number of points for a certain $x$-coordinate. For any $x$, the viable points are in the range $[-20 + |x|, 20 - |x|]$. This means that our total sum is equal to \begin{align*} 1 + 3 + 5 + \cdots + 41 + 39 + 37 + \cdots + 1 &= (1 + 3 + 5 + \cdots + 39) + (1 + 3 + 5 + \cdots + 41) \\ & = 20^2 + 21^2 \\ & = 29^2 \\ &= \boxed{\textbf{(C)} \, 841}. \end{align*}

考虑某个$x$坐标的点数。对于任意$x$，可行的点范围为$[-20 + |x|, 20 - |x|]$。这意味着总和等于 \begin{align*} 1 + 3 + 5 + \cdots + 41 + 39 + 37 + \cdots + 1 &= (1 + 3 + 5 + \cdots + 39) + (1 + 3 + 5 + \cdots + 41) \\ & = 20^2 + 21^2 \\ & = 29^2 \\ &= \boxed{\textbf{(C)} \, 841}. \end{align*}

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