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# Complex Numbers

"Complex" numbers share many properties with the real numbers. In fact, complex numbers include all the real numbers, so you know lots of them already. Such an overachiever.

\[ \large \begin{cases} {a_n=a_{n-1}a_{n-2}-b_{n-1}b_{n-2} } \\ { b_n=b_{n-1}a_{n-2}+a_{n-1}b_{n-2} } \end{cases} \]

Two sequences \(a_n\) and \(b_n\) begin with values \(a_1=b_1=a_2=b_2=1\) and satisfy the recursive relations above for \(n>2\).

Find \(\large \frac{b_{2015}}{a_{2015}}\).

If \(Z\) is a non-zero complex number, then find the minimum value of

\[\displaystyle \dfrac{\text{Im}(Z^{5})}{(\text{Im}(Z))^{5} }.\]

**Clarification:** \(\text{Im}(Z)\) represent the imaginary part of \(Z\).

For a complex number \(z\), find the smallest possible value of \[|z-3|^2+|z-5+2i|^2+|z-1+i|^2.\]

**Clarification:** \(i=\sqrt{-1}\).

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