Solubility of silver acetate in the presence of nitric acid
The process you are trying to investigate is:

$$\ce{AgOOCCH3(s) + HNO3(aq) &lt;=&gt; AgNO3 (aq) + CH3COOH (Aq)}$$

with the net ionic equation:

$$\ce{AgOOCCH3(s) + H+(aq) &lt;=&gt; Ag+(aq) + CH3COOH(aq)}$$

The equilibrium constant is

$$K_c = \dfrac{[\ce{Ag+}][\ce{CH3COOH}]}{[\ce{H+}]}$$

Note that this is law of mass action can be derived from the $K_a$ and the K_{sp}$ you have:

$$\begin{aligned}
K_{a}&amp;=\dfrac{[\ce{H+}][\ce{CH3COO-}]}{[\ce{CH3COOH}]}\\
K_{sp}&amp;=[\ce{Ag+}][\ce{CH3COO-}]\\
K_c &amp;= \dfrac{K_{sp}}{K_a}=\dfrac{[\ce{Ag+}][\ce{CH3COOH}]}{[\ce{H+}]}
\end{aligned}$$

The problem is now an ICE (Initial-change-equilibrium) problem. 

$$\begin{array}{|c|c|c|c|} \hline
\ &amp; [\ce{H+}] &amp; [\ce{Ag+}] &amp; [\ce{CH3COOH}]\\ \hline
I &amp; 0.1\ \mathrm{M} &amp; 0\ \mathrm{M} &amp; 0\ \mathrm{M}\\ 
C &amp; -x &amp; +x &amp; +x \\
E &amp; 0.1 -x \ \mathrm{M} &amp; x \ \mathrm{M} &amp; x \ \mathrm{M}\\ \hline
\end{array}$$

Plug values into the law of mass action, and solve for $x$, the concentration of $\ce{Ag+}$, and then you can determine the solubility.