1. Let $x,\,y,\,z>0$ and $a,\,b,\,c$ be real numbers such that $x^2+a^2=y^2+b^2=z^2+c^2=1$.

Prove that $(a+b+c)^2+(x+y+z)^2\ge 1$.

2. Originally Posted by anemone
Let $x,\,y,\,z>0$ and $a,\,b,\,c$ be real numbers such that $x^2+a^2=y^2+b^2=z^2+c^2=1$.

Prove that $(a+b+c)^2+(x+y+z)^2\ge 1$.

3. Originally Posted by Opalg

4. Originally Posted by Albert