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Polar Coordinates

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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