3. Let $O$ be the circumcenter of acute $\triangle A B C$, with $A B=6, A C=10$. If $\overrightarrow{A O}=x \overrightarrow{A B}+y \overrightarrow{A C}$, and $2 x+10 y=5$, then $\cos \angle B A C=$ $\qquad$ .
3. $\frac{1}{3}$.

From $\overrightarrow{A O} \cdot \overrightarrow{A C}=\frac{1}{2}|\overrightarrow{A C}|^{2}$, we get $60 x \cos \angle B A C+100 y=50$, then
$$
\cos \angle B A C=\frac{50-100 y}{60 x}=\frac{20 x}{60 x}=\frac{1}{3} .
$$