Competition problem — Combinatorics

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Combinatorics Difficulty 2.6 Junior Find the answer

The $\textit{taxicab distance}$ 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 $20$ ?

Pick one

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
$\begin{align*} 1 + 3 + 5 + \cdots + 41 + 39 + 37 + \cdots + 1 &= (1 + 3 + 5 + \cdots + 39) + (1 + 3 + 5 + \cdots + 41) \\ & = 20^2 + 21^2 \\ & = 29^2 \\ &= \boxed{\textbf{(C)} \, 841}. \end{align*}$
~mathboy100

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.