WakeUp wrote: The real numbers $a,b$ and $c$ satisfy the inequalities $|a|\ge |b+c|,|b|\ge |c+a|$ and $|c|\ge |a+b|$. Prove that $a+b+c=0$. By squaring and adding side by side them: $|a|\ge |b+c|,|b|\ge |c+a|$ and $|c|\ge |a+b|$ we get $(a+b+c)^2\le 0 \Longrightarrow \ \ \ a+b+c=0$

Let $ |b-c|\ge\sqrt{3}|a| , |c-a|\ge\sqrt{3}|b| , |a-b|\ge\sqrt{3}|c| ,$Prove that :$a+b+c=0$. http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&p=2703865 Let real numbers $a,b,c$ such that $\left| a-b \right|\ge \left| c \right|,\left| b-c \right|\ge \left| a \right|,\left| c-a \right|\ge \left| b \right|.$Prove that $a=b+c$ or $b=c+a$ or $c=a+b.$ Romania District Olympiad 2014 Moscow MO Grade 9 Problem 1

sqing wrote: Let $ |b-c|\ge\sqrt{3}|a| , |c-a|\ge\sqrt{3}|b| , |a-b|\ge\sqrt{3}|c| ,$Prove that :$a+b+c=0$. http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&p=2703865 Let real numbers $a,b,c$ such that $\left| a-b \right|\ge \left| c \right|,\left| b-c \right|\ge \left| a \right|,\left| c-a \right|\ge \left| b \right|.$Prove that $a=b+c$ or $b=c+a$ or $c=a+b.$ Romania District Olympiad 2014 Moscow MO Grade 9 Problem 1 https://artofproblemsolving.com/community/c6h579719p3421751 https://artofproblemsolving.com/community/c6h590111p3494061 https://artofproblemsolving.com/community/c6h1911572p13092028