3. In the Cartesian coordinate system $x O y$, $A$, $B$, and $C$ are three distinct points on the circle $x^{2}+y^{2}=1$. If there exist positive real numbers $\lambda, \mu$ such that $\overrightarrow{O C}=\lambda \overrightarrow{O A}+\mu \overrightarrow{O B}$, then the range of $(\lambda-1)^{2}+(\mu-3)^{2}$ is
3. $\left(\frac{1}{2},+\infty\right)$.

Let $\overrightarrow{O A_{1}}=\lambda \overrightarrow{O A}, \overrightarrow{O B_{1}}=\mu \overrightarrow{O B}$, and connect $B_{1} C, A_{1} C$.
Then $\left|\overrightarrow{O A_{1}}\right|=\lambda,\left|\overrightarrow{O B_{1}}\right|=\mu,|\overrightarrow{O C}|=1$.
Also, $\overrightarrow{O C}=\overrightarrow{O A_{1}}+\overrightarrow{O B_{1}}$, so $O, C, B_{1}$ form the three vertices of a triangle, and $\left|\overrightarrow{B_{1} C}\right|=\left|\overrightarrow{O A_{1}}\right|=\lambda$. Therefore, by the relationship between the sides of a triangle, we get
$$
\left\{\begin{array}{l}
\lambda+\mu>1, \\
|\lambda-\mu|<1.
\end{array}\right.
$$
Thus, $\lambda, \mu \in \left(\frac{1}{2},+\infty\right)$.