\documentclass[tikz,border={5pt 10pt 20pt 10pt}]{standalone}
\usetikzlibrary{patterns}
\usepackage{nccmath, mathtools}
\begin{document}
\begin{tikzpicture}[even odd rule]
% Calcul de sqrt(5)
\pgfmathsetmacro{\sqrtFive}{sqrt(5)}
% Définition des axes
\draw[thick,-latex] (-2,0) -- (5,0) node[right] {$x$}; % Axe des x
\draw[thick,-latex] (0,-3) -- (0,3) node[above] {$y$}; % Axe des y
% Régions
\begin{scope}
\fill[pattern=crosshatch dots, pattern color=purple] (1,0) circle (2) (1,0) circle (\sqrtFive);
\end{scope}
% Cercles
\draw[] (1,0) circle (2) (1,0) circle (\sqrtFive);
% Point z0
\coordinate (z0) at (1,0);
\node [below] at (z0) {$z_0$};
\fill (z0) circle [radius=1pt];
% Texte
\node at (4,3) {%
\(\medmath{%%
\begin{aligned}[t]
\left\lfloor x^2+y^2-2x \right\rfloor = 3
&\iff 3 \leq x^2+y^2-2x < 4 \\
&\iff 4 \leq \left( x - 1 \right) ^2 + y^2 < 5 \\
&\iff 2 \leq \sqrt{\left( x - 1 \right) ^2 + y^2} < \sqrt{5} \\
\end{aligned}}\)
};
% Texte
\node at (4,-2.7) {%
\(\medmath{%%
\text{Note : } \left\lfloor x \right\rfloor = n \iff n \leq x < n+1
}%%
\)
};%
\end{tikzpicture}
\end{document}
