If a, b and c are three integers such that at least two of them are unequal and `omega (ne 1)` is a cube root of unit, then the least value of the expression `|a + b omega + c omega^(2)|` is

Options

A. 0
B. 1
C. `(sqrt(3))/(2)`
D. `(1)/(2)`

Answer

To find the least value of the expression \( |a + b \omega + c \omega^2| \), where \( \omega \) is a cube root of unity and at least two of the integers \( a, b, c \) are unequal, we can follow these steps:

### Step 1: Understand the properties of \( \omega \)
The cube roots of unity are given by:
\[ \omega = e^{2\pi i / 3} = -\frac{1}{2} + i \frac{\sqrt{3}}{2} \]
and
\[ \omega^2 = e^{-2\pi i / 3} = -\frac{1}{2} - i \frac{\sqrt{3}}{2} \]
We also know that:
\[ 1 + \omega + \omega^2 = 0 \]

### Step 2: Rewrite the expression
We can express \( \omega^2 \) in terms of \( \omega \):
\[ \omega^2 = -1 - \omega \]
Thus, we can rewrite the expression:
\[ |a + b \omega + c \omega^2| = |a + b \omega + c(-1 - \omega)| = |(a - c) + (b - c) \omega| \]

### Step 3: Identify the real and imaginary parts
Let \( x = a - c \) and \( y = b - c \). Then the expression simplifies to:
\[ |x + y \omega| = |x + y\left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right)| \]
This gives us:
\[ |x - \frac{y}{2} + i \frac{\sqrt{3}}{2} y| \]

### Step 4: Calculate the modulus
The modulus of a complex number \( z = x + iy \) is given by:
\[ |z| = \sqrt{x^2 + y^2} \]
Thus, we have:
\[ |x - \frac{y}{2} + i \frac{\sqrt{3}}{2} y| = \sqrt{\left(x - \frac{y}{2}\right)^2 + \left(\frac{\sqrt{3}}{2} y\right)^2} \]

### Step 5: Expand and simplify
Expanding this, we get:
\[ = \sqrt{\left(x - \frac{y}{2}\right)^2 + \frac{3}{4} y^2} \]
\[ = \sqrt{x^2 - xy + \frac{1}{4}y^2 + \frac{3}{4}y^2} \]
\[ = \sqrt{x^2 - xy + y^2} \]

### Step 6: Find the minimum value
To minimize \( \sqrt{x^2 - xy + y^2} \), we note that since \( a, b, c \) are integers and at least two are unequal, the minimum distance between any two integers is 1. Therefore, we can set \( |y| = 1 \) and \( |x| = 1 \), leading to:
\[ x^2 - xy + y^2 \geq 1 \]
The minimum value occurs when \( x = 1 \) and \( y = 1 \) or any permutation thereof, yielding:
\[ \sqrt{1^2 - 1 \cdot 1 + 1^2} = \sqrt{1} = 1 \]

### Conclusion
Thus, the least value of the expression \( |a + b \omega + c \omega^2| \) is:
\[ \boxed{1} \]