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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.

Level 4

         

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

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

How many complex numbers \(z\) are there such that \(|z| = 1\) and \[\large{\left|\dfrac{z}{\bar{z}}+\dfrac{\bar{z}}{z}\right|} =1?\]

Let \( z=a+bi\), \(|z|=5\), and \(b>0\). If the distance between \((1+2i)z^3\) and \(z^5\) is as large as possible, then determine \(z^4\). Give your answer as the sum of the real and imaginary parts of \(z^4\).

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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