Question

Let \( \omega (\neq 1) \) be a cubic root of unity. Then the minimum value of the set \( \{ |a + b\omega + c\omega^2|^2 : a, b, c \) are distinct non-zero integers \( \} \) equals:

• A. \( 15 \)
• B. \( 5 \)
• C. \( 3 \)
• D. \( 4 \)

Hint:

Remember the fundamental properties of cubic roots of unity, especially \( 1 + \omega + \omega^2 = 0 \) and \( \omega^3 = 1 \), as they are crucial for simplifying expressions involving \( \omega \).

Answer 1

The Correct Option is C

Step 1: Recall the properties of cubic roots of unity.
The cubic roots of unity are \( 1, \omega, \omega^2 \), where \( \omega = e^{i2\pi/3} = -\frac{1}{2} + i\frac{\sqrt{3}}{2} \) and \( \omega^2 = e^{i4\pi/3} = -\frac{1}{2} - i\frac{\sqrt{3}}{2} \). These roots satisfy the following properties: % Option (A) \( 1 + \omega + \omega^2 = 0 \) % Option (B) \( \omega^3 = 1 \)
Step 2: Simplify the expression \( |a + b\omega + c\omega^2|^2 \).
We know that \( |z|^2 = z \bar{z} \), where \( \bar{z} \) is the complex conjugate of \( z \). The conjugate of \( \omega \) is \( \bar{\omega} = \omega^2 \), and the conjugate of \( \omega^2 \) is \( \bar{\omega}^2 = \omega \).
\begin{align} |a + b\omega + c\omega^2|^2 &= (a + b\omega + c\omega^2)(a + b\bar{\omega} + c\bar{\omega}^2)
&= (a + b\omega + c\omega^2)(a + b\omega^2 + c\omega)
&= a^2 + ab\omega^2 + ac\omega + ba\omega + b^2\omega^3 + bc\omega^2 + ca\omega^2 + cb\omega^4 + c^2\omega^3
&= a^2 + b^2(1) + c^2(1) + ab(\omega + \omega^2) + ac(\omega + \omega^2) + bc(\omega^2 + \omega^4) \end{align} Using \( 1 + \omega + \omega^2 = 0 \), we have \( \omega + \omega^2 = -1 \). Also, \( \omega^4 = \omega^3 \cdot \omega = 1 \cdot \omega = \omega \), so \( \omega^2 + \omega^4 = \omega^2 + \omega = -1 \).
\begin{align} |a + b\omega + c\omega^2|^2 &= a^2 + b^2 + c^2 + ab(-1) + ac(-1) + bc(-1)
&= a^2 + b^2 + c^2 - ab - ac - bc
&= \frac{1}{2} (2a^2 + 2b^2 + 2c^2 - 2ab - 2ac - 2bc)
&= \frac{1}{2} ((a^2 - 2ab + b^2) + (a^2 - 2ac + c^2) + (b^2 - 2bc + c^2))
&= \frac{1}{2} ((a - b)^2 + (a - c)^2 + (b - c)^2) \end{align}
Step 3: Find the minimum value of \( \frac{1}{2} ((a - b)^2 + (a - c)^2 + (b - c)^2) \) where \( a, b, c \) are distinct non-zero integers.
To minimize this expression, we need to choose distinct non-zero integers \( a, b, c \) such that the squares of their differences are as small as possible. The smallest possible absolute differences between three distinct integers are \( |a - b| = 1 \), \( |b - c| = 1 \) (which implies \( |a - c| = 2 \)), or permutations thereof.
Let \( \{|a - b|, |a - c|, |b - c|\} = \{1, 1, 2\} \). The squares are \( 1, 1, 4 \).
The minimum value is \( \frac{1}{2} (1^2 + 1^2 + 2^2) = \frac{1}{2} (1 + 1 + 4) = \frac{6}{2} = 3 \). This can be achieved with \( a=1, b=2, c=3 \) or any permutation with differences of 1 and 2.