그림과 같이 삼각형 $\mathrm{OAB}$ 의 무게중심을 $\mathrm{G}$, 선분 $\mathrm{AB}$ 를 $1: 5$ 로 내분하는 점을 $\mathrm{C}$ 라 하자. $\overrightarrow{\mathrm{OA}}=\vec{a}, \overrightarrow{\mathrm{OB}}=\vec{b}$ 라 할 때, 두 실수 $m, n$ 에 대하여 $\overrightarrow{\mathrm{GC}}=m \vec{a}+n \vec{b}$ 이다. $m+n$ 의 값은?
(1) $\frac{1}{6}$
(2) $\frac{1}{3}$
(3) $\frac{1}{2}$
(4) $\frac{2}{3}$
(5) $\frac{5}{6}$

Step 1 ：Let G be the centroid of triangle OAB, and C be a point on line segment AB such that AC:CB = 1:5. We need to find the values of m and n such that \(\overrightarrow{GC} = m\vec{a} + n\vec{b}\).

Step 2 ：Find the position vector of C: \(\overrightarrow{OC} = \frac{5\vec{a} + 1\vec{b}}{1+5}\)

Step 3 ：Find the position vector of G: \(\overrightarrow{OG} = \frac{\vec{a} + \vec{b}}{3}\)

Step 4 ：Find the vector \(\overrightarrow{GC}\): \(\overrightarrow{GC} = \overrightarrow{OC} - \overrightarrow{OG}\)

Step 5 ：Substitute the expressions for \(\overrightarrow{OC}\) and \(\overrightarrow{OG}\): \(\overrightarrow{GC} = \frac{5\vec{a} + \vec{b}}{6} - \frac{\vec{a} + \vec{b}}{3}\)

Step 6 ：Simplify the expression: \(\overrightarrow{GC} = \frac{3\vec{a} - \vec{b}}{6}\)

Step 7 ：Compare this expression with \(m\vec{a} + n\vec{b}\) to find the values of m and n: \(m = \frac{1}{2}\) and \(n = -\frac{1}{6}\)

Step 8 ：Calculate the value of \(m+n\): \(m+n = \frac{1}{2} - \frac{1}{6}\)

Step 9 ：\(\boxed{\frac{1}{3}}\) is the final answer.