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\begin{document}

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\begin{center}
{\Large NMU Math \& CS Department} \\
\vspace{0.5cm}
{\Large Problem of the Month, March 2023}
\end{center}

\vspace{1cm}

\paragraph{} Draw a quarter circle of radius one centered at $(0,0)$. Then draw a quarter circle centered at $(1,1)$ with appropriate radius so that it is tangent to the first quarter circle.

\paragraph{} Find the exact value of the radius of the small circle (in the form $r+s\sqrt{d}$ where $r,s \in \mathbb{Q}\hspace{2pt}, d \in \mathbb{N}$) that is tangent to both quarter circles and the vertical line, as illustrated.

\vspace{0.5cm}

\begin{center}
\begin{tikzpicture}[scale=2] 
\coordinate (A) at (0,0);
\node at (A) [left=1mm of A] {$(0,0)$};
\coordinate (B) at (5,0);
\node at (B) [right=1mm of B] {$(1,0)$};
\coordinate (C) at (5,5);
\node at (C) [right=1mm of C] {$(1,1)$};
\coordinate (D) at (0,5);
\node at (D) [left=1mm of D] {$(0,1)$};
\draw[thick] (5,0) arc (0:90:5cm);
\draw[thick] (2.94,5) arc (180:270:2.05cm);
\filldraw[fill=black!3,thick](4.66,2.625) circle (0.33);
\draw (A) -- (B) -- (C) -- (D) -- (A);
\end{tikzpicture}
\end{center}

\end{document}