Placing an object at the center of curvature of a concave shaving mirror is like holding your face at twice the focal length where you see a real, inverted image of the exact same size right back at you.

Remember 'F-C-F': Object at $C$ ($2f$) forms an image at $C$ ($2f$).

According to the mirror formula, $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$, where $f$ is the focal length, $v$ is the image distance, and $u$ is the object distance. Using the New Cartesian Sign Convention for a concave mirror, the focal length $f = -10\text{ cm}$ and the real image distance $v = -20\text{ cm}$. Substituting these values into the mirror formula gives $\frac{1}{-10} = \frac{1}{-20} + \frac{1}{u}$, which yields $\frac{1}{u} = -\frac{1}{10} + \frac{1}{20} = -\frac{1}{20}$, resulting in $u = -20\text{ cm}$.