<s>[INST] Show that if $a+b+c=0$, $2(a^4 + b^4+ c^4)$ is a perfect square 
Show that for $\{a,b,c\}\subset\Bbb Z$ if $a+b+c=0$ then $2(a^4 + b^4+ c^4)$ is a perfect square. 

This question is from a math olympiad contest. 
I started developing the expression $(a^2+b^2+c^2)^2=a^4+b^4+c^4+2a^2b^2+2a^2c^2+2b^2c^2$ but was not able to find any useful direction after that.
Note: After getting 6 answers here, another user pointed out other question in the site with similar but not identical content (see above), but the 7 answers presented include more comprehensive approaches to similar problems (e.g. newton identities and other methods) that I found more useful, as compared with the 3 answers provided to the other question. 
 [/INST] We can simplify the algebra by noticing that the expressions are homogenous:
$
2(a^4+b^4+c^4)
\\\quad=2(a^4+b^4+(a+b)^4)
\\\quad=2b^4(x^4+1+(x+1)^4),
\quad x=a/b
$
$
2(x^4+1+(x+1)^4)
\\\quad=4 x^4 + 8 x^3 + 12 x^2 + 8 x + 4
\\\quad=4 (x^2 + x)^2 + 8 (x^2 + x) + 4
\\\quad=4(y^2+2y+1),
\quad y=x^2+x
\\\quad=4(y+1)^2
\\\quad=(2(y+1))^2
\\\quad=(2(x^2+x+1))^2
$
$
2(a^4+b^4+c^4)
\\\quad=b^4(2(x^2+x+1))^2
\\\quad=(2b^2(x^2+x+1))^2
\\\quad=(2(a^2+ab+b^2)^2
$
</s>