KöMaL 2005 A371

Given that $a$+$b \le c$+1, $b$+$c$ $\le a$+1, $c$+$a \le b$+1 for the numbers $a$,$b$,$c$ $\ge$0, show that $a^2$+$b^2$+$c^2 \le$2 abc +1.