Algebra

Complex Numbers

Complex Numbers: Level 4 Challenges

         

{an=an1an2bn1bn2bn=bn1an2+an1bn2 \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 ana_n and bnb_n begin with values a1=b1=a2=b2=1a_1=b_1=a_2=b_2=1 and satisfy the recursive relations above for n>2n>2.

Find b2015a2015\large \frac{b_{2015}}{a_{2015}}.

How many complex numbers zz are there such that z=1|z| = 1 and zzˉ+zˉz=1?\large{\left|\dfrac{z}{\bar{z}}+\dfrac{\bar{z}}{z}\right|} =1?

Let z=a+bi z=a+bi, z=5|z|=5, and b>0b>0. If the distance between (1+2i)z3(1+2i)z^3 and z5z^5 is as large as possible, then determine z4z^4. Give your answer as the sum of the real and imaginary parts of z4z^4.

If ZZ is a non-zero complex number, then find the minimum value of

Im(Z5)(Im(Z))5.\displaystyle \dfrac{\text{Im}(Z^{5})}{(\text{Im}(Z))^{5} }.

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

For a complex number zz, find the smallest possible value of z32+z5+2i2+z1+i2.|z-3|^2+|z-5+2i|^2+|z-1+i|^2.

Clarification: i=1i=\sqrt{-1}.

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