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Initial mathematical cohesion problem
Known a+b+c=0 a? +b? +c? = 1 so (a+b+c)? =0

Answer? +b? +c? +2 (AB+AC+BC) = 01+2 (AB+AC+BC) = 0 so ab+ac+bc=- 1/2 (ab+ac+bc)? = 1/4, that is, a? b? +a? c? +b? c? +2(a? bc+b? ac+c? ab)= 1/4 a? b? +a? c? +b? c? +2abc(a+b+c)= 1/4 (called a+b+c=0)

So a? b? +a? c? +b? c? = 1/4,

(a? +b? +c? )? =a^4+b^4+c^4+2 a? b? +2a? c? +2b? c?

=a^4+b^4+c^4+2(a? b? +a? c? +b? c? )

So a 4+b 4+c 4 = (a? +b? +c? )? - 2(a? b? +a? c? +b? c? )= 1- 2*( 1/4)= 1-( 1/2)= 1/2

(a? +b? +c? = 1,a? b? +a? c? +b? c? = 1/4 )

The final answer is 1/2.