Algebra

Polar Coordinates

Polar Coordinates: Level 4 Challenges

         

How many complex numbers a+bia + bi are there where aa and bb are integers and

a+bi5? |a + bi| \leq 5 ?

Details and assumptions:
a+bi |a+bi| denotes the modulus or absolute value.

{56x+33y=yx2+y233x56y=xx2+y2\large\begin{cases} {56x +33y = -\frac y{x^2+y^2} } \\ {33x -56y = \frac x{x^2+y^2} } \end{cases}

Given that x,yx,y are complex numbers that satisfy the system of equations above and that x+y |x| + |y| equals pq\frac pq for coprime positive integers p,qp,q, evaluate 6pq6p-q.

Let z1=6+iz_1 = 6+i and z2=43iz_2 = 4-3i.
Let zz be a complex number such that arg(zz1z2z)=π2\text{arg}\left(\dfrac{z-z_1}{z_2-z}\right)=\dfrac{\pi}{2}, and z(5i)|z-(5-i)|=m\sqrt{m}.

Find m.m.


Details and Assumptions:

  • arg(x) \text{arg}(x) is the argument of xx.
  • z1,z2,z_1,z_2, and zz are complex numbers.

Let xx be a solution to the equation x2+x+2014=0x^2+x+2014=0.

Find the value of limni=12nxi2014i12+1n.\lim\limits_{n\to \infty} \sqrt[\Large n]{\prod_{i=1}^{2n}\left|\dfrac{x^i}{2014^{\frac{i-1}{2}}}+1\right|}.

Let z1 z_{1} and z2 z_{2} be the two complex roots of the equation z2+az+b=0 z^{2} +az+ b =0 , where aa and bb are real numbers. Further, assume that the origin, z1 z_{1} and z2 z_{2} form an equilateral triangle. Then:

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