This is an Olympiad question.
In a $\triangle ABC$, where $\sin A+\sin B+\sin C\leq 1$, prove that one of the angles is more than $150^\circ$.
First of all I assumed that WLOG, $A\geq B \geq C$. Then I tried solving the problem using triangle inequality and Sine rule. From there, and from the given statement in the question, I was able to establish that $\sin A < 1/2$.
I know that I just need to establish that angle $A$ is greater than $90^\circ$ or something like that.