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# Polar Coordinates

Polar coordinates are a way to describe where a point is on a plane. Instead of using x and y, you use the angle theta and radius r, to describe the angle and distance of the point from the origin.

# Polar Coordinates: Level 4 Challenges

How many complex numbers $$a + bi$$ are there where $$a$$ and $$b$$ are integers and

$|a + bi| \leq 5 ?$

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

$\large\begin{cases} {56x +33y = -\frac y{x^2+y^2} } \\ {33x -56y = \frac x{x^2+y^2} } \end{cases}$

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

Let $$z_1 = 6+i$$ and $$z_2 = 4-3i$$. Let $$z$$ be a complex number such that $$\text{arg}\left(\dfrac{z-z_1}{z_2-z}\right)=\dfrac{\pi}{2}$$, and $$|z-(5-i)|$$=$$\sqrt{m}$$. Find $$m.$$

Details and Assumptions:

• $$\text{arg}(x)$$ is the argument of $$x$$.

• $$z_1,z_2,$$ and $$z$$ are complex numbers.

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

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

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

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