Consider the complex numbers $x,y,z$ such that
$|x|=|y|=|z|=1$. Define the number
$$a=\left (1+\frac xy\right )\left (1+\frac yz\right )\left (1+\frac zx\right ).$$
$\textbf{(a)}$ Prove that $a$ is a real number.
$\textbf{(b)}$ Find the minimal and maximal value $a$ can achieve, when $x,y,z$ vary subject to $|x|=|y|=|z|=1$.

[i] (Stefan Bălăucă & Vlad Robu)[/i]
### Part (a): Prove that \( a \) is a real number

Given the complex numbers \( x, y, z \) such that \( |x| = |y| = |z| = 1 \), we need to show that the expression 
\[ a = \left(1 + \frac{x}{y}\right)\left(1 + \frac{y}{z}\right)\left(1 + \frac{z}{x}\right) \]
is a real number.

1. **Express the modulus condition**:
   Since \( |x| = |y| = |z| = 1 \), we have \( x \overline{x} = y \overline{y} = z \overline{z} = 1 \).

2. **Simplify the expression**:
   Consider the product:
   \[
   a = \left(1 + \frac{x}{y}\right)\left(1 + \frac{y}{z}\right)\left(1 + \frac{z}{x}\right)
   \]

3. **Use the property of complex conjugates**:
   Since \( |x| = |y| = |z| = 1 \), we can write \( \frac{x}{y} \overline{\left(\frac{x}{y}\right)} = 1 \), \( \frac{y}{z} \overline{\left(\frac{y}{z}\right)} = 1 \), and \( \frac{z}{x} \overline{\left(\frac{z}{x}\right)} = 1 \).

4. **Expand the product**:
   \[
   a = \left(1 + \frac{x}{y}\right)\left(1 + \frac{y}{z}\right)\left(1 + \frac{z}{x}\right)
   \]
   Expanding this product, we get:
   \[
   a = 1 + \frac{x}{y} + \frac{y}{z} + \frac{z}{x} + \frac{x}{y} \cdot \frac{y}{z} + \frac{y}{z} \cdot \frac{z}{x} + \frac{z}{x} \cdot \frac{x}{y} + \frac{x}{y} \cdot \frac{y}{z} \cdot \frac{z}{x}
   \]

5. **Simplify using the fact that \( \frac{x}{y} \cdot \frac{y}{z} \cdot \frac{z}{x} = 1 \)**:
   \[
   a = 1 + \frac{x}{y} + \frac{y}{z} + \frac{z}{x} + \frac{x}{z} + \frac{y}{x} + \frac{z}{y} + 1
   \]
   \[
   a = 2 + \frac{x}{y} + \frac{y}{z} + \frac{z}{x} + \frac{x}{z} + \frac{y}{x} + \frac{z}{y}
   \]

6. **Combine terms**:
   Notice that each term and its conjugate appear in the sum:
   \[
   a = 2 + \left(\frac{x}{y} + \frac{y}{x}\right) + \left(\frac{y}{z} + \frac{z}{y}\right) + \left(\frac{z}{x} + \frac{x}{z}\right)
   \]

7. **Use the fact that \( \frac{x}{y} + \frac{y}{x} \) is real**:
   Since \( \frac{x}{y} + \frac{y}{x} \) is the sum of a complex number and its conjugate, it is real. Similarly, \( \frac{y}{z} + \frac{z}{y} \) and \( \frac{z}{x} + \frac{x}{z} \) are real.

8. **Conclude that \( a \) is real**:
   Therefore, \( a \) is the sum of real numbers and hence is itself real.

\(\blacksquare\)

### Part (b): Find the minimal and maximal value \( a \) can achieve

Given \( a = \left(1 + \frac{x}{y}\right)\left(1 + \frac{y}{z}\right)\left(1 + \frac{z}{x}\right) \), we need to find the range of \( a \).

1. **Express \( a \) in terms of sums**:
   Using the fact that \( |x| = |y| = |z| = 1 \), we have:
   \[
   a = 2 + x \overline{y} + y \overline{z} + z \overline{x} + x \overline{z} + y \overline{x} + z \overline{y}
   \]

2. **Rewrite using the sum of complex numbers**:
   Let \( S = x + y + z \). Then:
   \[
   a = (x + y + z)(\overline{x} + \overline{y} + \overline{z}) - 1 = |S|^2 - 1
   \]

3. **Determine the range of \( |S|^2 \)**:
   Since \( |x| = |y| = |z| = 1 \), the maximum value of \( |S| \) occurs when \( x, y, z \) are all the same, giving \( |S| = 3 \). The minimum value of \( |S| \) occurs when \( x, y, z \) are such that their sum is zero, giving \( |S| = 0 \).

4. **Calculate the range of \( a \)**:
   \[
   \text{If } |S| = 3, \text{ then } a = 3^2 - 1 = 8
   \]
   \[
   \text{If } |S| = 0, \text{ then } a = 0^2 - 1 = -1
   \]

5. **Verify the extremal values**:
   - For \( a = 8 \): Let \( x = y = z = 1 \). Then \( |x + y + z| = 3 \) and \( a = 8 \).
   - For \( a = -1 \): Let \( x = 1 \), \( y = e^{2\pi i / 3} \), \( z = e^{4\pi i / 3} \). Then \( x + y + z = 0 \) and \( a = -1 \).

Thus, the minimal and maximal values of \( a \) are \(-1\) and \(8\), respectively.

\(\boxed{-1 \leq a \leq 8}\)