Question 3
If $$a+b+c=0$$, then the value of $$\frac{(a^2+b^2+c^2)^2}{a^4+b^4+c^4}$$ is
Solution
Let $$S=a^2+b^2+c^2$$. Since $$a+b+c=0$$, we have $$ab+bc+ca=-\frac{S}{2}$$ and $$a^2b^2+b^2c^2+c^2a^2=(ab+bc+ca)^2=\frac{S^2}{4}$$. Thus $$a^4+b^4+c^4=S^2-2\left(\frac{S^2}{4}\right)=\frac{S^2}{2}$$, so the required ratio is $$2$$.
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