Algebra

# 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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