Complex Numbers

Complex Numbers: Level 4 Challenges


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

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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