Symbolic Operators

\[x * y = \frac{1}{x} + \frac{1}{y}, \quad x \# y = \frac{x+y}{x-y}\]

Let the operations \(\#\) and \(*\) be defined as described above.

Find the value of \(k\) such that

\[(22 * k) \# (k * 33) = 27.\]

Suppose we define the function \(@\) such that \(a @ b = 3a - 2b + a b\).

What is the value of \(10@4 - 4@10?\)

If \( a \oplus b = \left( a+b \right) ^{ 2 } \) , what is the value of \( 3\oplus 1\)?

Consider the arithmetic operations \(\blacktriangle\) and \(\blacktriangledown\) defined by

\[\large \begin{align} a \blacktriangle b & = \begin{cases} a & \text{ if } \lvert a \rvert \geq \lvert b \rvert \\ b & \text{ if } \lvert a \rvert < \lvert b \rvert \end{cases} \\ a\blacktriangledown b & = \begin{cases} a & \text{ if } \lvert a \rvert \leq \lvert b \rvert \\ b & \text{ if } \lvert a \rvert > \lvert b \rvert \end{cases} \end{align} \]

How many integers \(k\) are there such that \((-20 \blacktriangle 9) \blacktriangledown (k \blacktriangle 5)=5?\)

For any two real numbers \(a\) and \(b\), the operation \(\oplus\) defined by \(a\oplus b=ab+1\) is \(\text{____________________}.\)