KG-MO 2009 Problem 7

Does $a^2 + b^2 + c^2 \leqslant 2(ab + bc + ca)$ hold for every $a,b,c$ if it is known that $a^4 + b^4 + c^4 \leqslant 2(a^2 b^2 + b^2 c^2 + c^2 a^2 )$ .