\section*{Problem 6 - 061046}

Give the total number of all different integer solution pairs \((x, y)\) of the inequality

\[
|x|+|y| \leq 100
\]

where two solution pairs \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\) are considered the same if and only if \(x_{1}=x_{2}\) and \(y_{1}=y_{2}\).
}

If \(x=0\), then for \(y\) the possible values are: \(-100 \ldots 100 \Rightarrow\) Number: 201

If \(x=-1\) or 1, then for \(y\) the possible values are: \(-99 \ldots 99 \Rightarrow\) Number: \(2 \cdot 199\)

If \(x=-2\) or 2, then for \(y\) the possible values are: \(-98 \ldots 98 \Rightarrow\) Number: \(2 \cdot 197\)

and so on until

If \(x=-99\) or 99, then for \(y\) the possible values are: \(-1 \ldots 1 \Rightarrow\) Number: \(2 \cdot 3\)

If \(x=-100\) or 100, then for \(y\) the possible values are: \(0 \Rightarrow\) Number: \(2 \cdot 1\)

\[
2 \sum_{i=1}^{100}(2 i-1)+201=2 \cdot 100^{2}+201=20201
\]

Solution by Matthias Lösche

\subsection*{7.9 VII. Olympiad 1967}

\subsection*{7.9.1 First Round 1967, Grade 10}