2016/115) For $x,\,y$ and $z\in [0,\,1]$ such that $xy+yz+zx=1$, prove x,y.z can be sides of a triangle.

letting $x=\tan\,A$, $y=\tan\, B$, $z\tan\, C$ we have A, B, C, between 0 and $\frac{\pi}{4}$
also $\tan\, A \tan\,  B +  \tan\, B \tan\,  C + \tan\, C \tan\,  A = 1$
using $\tan(A+B+C)$ we get
$A+B+C=\frac{\pi}{2}$
so $B+C>=A$ from above and A between 0 and $\frac{\pi}{4}$
so $x <= y+z$
similarly $y <= z + x$ and $z <= x + y$ so x,y,z can be sides of a triangle