Given: $|\overrightarrow a|=|\overrightarrow b|=|\overrightarrow c|=|\overrightarrow a+\overrightarrow b+\overrightarrow c|=1$
$|\overrightarrow a+\overrightarrow b+\overrightarrow c|^2=|\overrightarrow a|^2+|\overrightarrow b|^2+|\overrightarrow c|^2+2(\overrightarrow a.\overrightarrow b+\overrightarrow b.\overrightarrow c+\overrightarrow c.\overrightarrow a)$
$\Rightarrow\:1=3+2(cos\alpha+cos\beta+cos\gamma)$
$\Rightarrow\:cos\alpha+cos\beta+cos\gamma=-1$
$\Rightarrow\:$ atleast one out of $cos\alpha,\:cos\beta,\:cos\gamma$ should be $ -ve$.
$\Rightarrow\:$ At least one out of the three angles should be obtuse.