$$\eqalign{
& a)\,\,\overrightarrow {AM} = {1 \over 2}\overrightarrow {AO} + {1 \over 2}\overrightarrow {AB} \cr
& = {1 \over 2}\overrightarrow {AO} + {1 \over 2}\left( {\overrightarrow {AO} + \overrightarrow {OB} } \right) \cr
& = \overrightarrow {AO} + {1 \over 2}\overrightarrow {OB} \cr
& = {1 \over 2}\overrightarrow {OB} - \overrightarrow {OA} \cr
& b)\,\,\overrightarrow {BN} = {1 \over 2}\left( {\overrightarrow {BO} + \overrightarrow {BC} } \right) \cr
& = {1 \over 2}\overrightarrow {BO} + {1 \over 2}\left( {\overrightarrow {BO} + \overrightarrow {OC} } \right) \cr
& = \overrightarrow {BO} + {1 \over 2}\overrightarrow {OC} \cr
& = {1 \over 2}\overrightarrow {OC} - \overrightarrow {OB} \cr
& c)\,\,\overrightarrow {MN} = {1 \over 2}\overrightarrow {BC} = {1 \over 2}\left( {\overrightarrow {OC} - \overrightarrow {OB} } \right) \cr} $$
