For $a,b,c \in \mathbb{R}$ and $|a| \geq |b+c|, |b| \geq |c+a|, |c| \geq |a+b|$ prove then $a+b+c = 0$ 
For $a,b,c \in \mathbb{R}$ and $|a| \geq |b+c|, |b| \geq |c+a|, |c|
 \geq |a+b|$ prove then $a+b+c = 0$


Solution:
$$
|a| + |b| + |c| \geq |b+c| + |c+a| + |a+b| \geq 0
$$
and
$$0 \leq |a+b+c|\leq |a+b| + |c| \leq |a| + |b| + |c|$$
so we have 
$$0 \leq |a+b+c| \geq 0 $$
$$ |a+b+c|= 0 \iff a+b+c =0 $$ 
I need check my solution. 
Many thanks!

hint: Square each of the $| ..|$  inequality and consider $f(a) = a^2 + (2b+2c)a + (b+c)^2$. Show that $f(a) \ge 0$. 
