"Complex" numbers share many properties with the real numbers. In fact, complex numbers include all the real numbers, so you know lots of them already. Such an overachiever.

The expression \( \displaystyle \frac{1}{2-11i} \) can be rewritten with a real denominator (using the complex conjugate) in the form \[ \displaystyle \frac{a+bi}{c}, \]

where \( a \), \( b \), and \( c \) are positive, co-prime integers. What is the sum of \( a \), \( b \), and \( c \)?

**Details and assumptions**

\( i \) is the the imaginary unit, defined by \( i^2 = -1 \).

Consider the complex numbers \[A=12+i \mbox{ and } B=11-3i.\] If the value of \(2A\overline{A}+2\overline{A}B+A\overline{B}+B\overline{B}\) is \(X+Yi\), what is the value of \(X-Y\)?

**Details and assumptions**

\(\overline{A}\) denotes the complex conjugate of the complex number \(A\).

\(i\) is the imaginary number that satisfies \(i^2 = -1\).

For complex numbers \(\alpha=5+7i\) and \(\beta=3-i,\) what is the value of \[\alpha \overline{\alpha} +\overline{\alpha} \beta +\alpha \overline{\beta} +\beta \overline{\beta}?\]

**Details and assumptions**

\(\overline{\alpha}\) denotes the complex conjugate of the complex number \(\alpha\).

\(i\) is the imaginary number that satisfies \(i^2 = -1\).

Consider the complex numbers \[A=8-2i \mbox{ and } B=4-i.\] What is the value of \((A+B)^2+\left(\overline{A+B}\right)^2\)?

**Details and assumptions**

\(\overline{A}\) denotes the complex conjugate of the complex number \(A\).

\(i\) is the imaginary number that satisfies \(i^2 = -1\).

Let \(\omega\) and \(z\) be the complex numbers \[ \omega =2-i, \quad z =\frac{\omega+4}{4\omega-1}. \]

If \(z\overline{z}\) can be expressed as \(\frac{a}{b},\) where \(a\) and \(b\) are coprime positive integers, what is \(a+b?\)

**Details and assumptions**

\(\overline{z}\) denotes the complex conjugate of the complex number \(z\).

\(i\) is the imaginary number that satisfies \(i^2 = -1\).

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