I think that there is probably some issue with the problem as it is stated, because we can show that a=b=c=0, hence R is undefined.
Here's an approach using the converse of the Cauchy-Schwarz inequality. Take the vector u = (a,b,c) and the vector v = (1,1,1). We will show that v.w (dot product) is equal to ||v||*||w|| (multiply two norms). This can only happen if one vector is a multiple of the other.
In fact its a bit easier, given the information we have, to square both sides.
The dot product squared is ((1,1,1).(a,b,c))=(a+b+c)^2. The product of the norms squared is (1^2+1^2+1^2)*(a^2+b^2+c^2) = 3(a^2+b^2+c^2).
The problem statement tells us that the dot product squared equals the product of the norms squared. The Cauchy-Schwarz inequality says that these can only agree if (a,b,c) is a multiple of (1,1,1). So there is a scalar t so that (a,b,c)=t*(1,1,1)=(t,t,t). Thus a=b=c=t.
Now getting back to the question: (a+b+c)^2=(3a)^2 = 9a^2, and we are told this equals 3a^2. The only way this can be true is if a^2 = 0. Hence a = b= c = 0.
So the expression (a^2+b^2+c^2)/(a^4+b^4+c^4) is undefined.
