Let ω ≠ 1 be a cube root of unity. Then the minimum of the set { |a + bω + cω²|² : a, b, c distinct

Q: Wait, why can't the minimum be 0? If $a=b=c$, the expression becomes 0, right?

You are absolutely right that if $a=b=c$, the expression evaluates to zero. However, look closely at the problem statement: '$a, b, c$ are distinct non-zero integers.' The word 'distinct' is the trap! It forbids $a=b=c$. Therefore, the differences $(a-b)$, $(b-c)$, and $(c-a)$ can never be zero. We must find the smallest non-zero integer values for these differences.

Q: Does the order of $a, b, c$ matter? What if I chose $a=3, b=2, c=1$ instead of $1, 2, 3$?

That is a brilliant question. Because the expression $E = \frac{1}{2}[(a-b)^2 + (b-c)^2 + (c-a)^2]$ is symmetric with respect to $a, b,$ and $c$, the order does not change the final value. Whether you choose $(1, 2, 3)$ or $(3, 2, 1)$, the squared differences remain the same. The symmetry is the beauty of this algebraic form!

Q: Why did we use the conjugate property $|z|^2 = z \cdot \bar{z}$? Is there a faster way?

While you could try to substitute $\omega = -\frac{1}{2} + i\frac{\sqrt{3}}{2}$, that path leads to a messy expansion with real and imaginary parts. Using the conjugate property is the 'JEE way'—it is elegant, minimizes calculation errors, and leverages the inherent properties of complex numbers. It transforms a complex problem into a purely real algebraic one.

Question

Let $\omega 
e 1$ be a cube root of unity. Then the minimum of the set $\{ |a + b\omega + c\omega^2|^2 : a, b, c 	ext{ distinct non-zero integers} \}$ equals

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, future engineers. Today, we are not just solving a problem; we are peeling back the layers of a beautiful algebraic structure. We are looking at the expression $E = |a + b\omega + c\omega^2|^2$, where $\omega$ is a complex cube root of unity.

At first glance, this looks like a daunting task involving complex numbers, but I want you to take a deep breath. In the world of JEE Advanced, complexity is often just a mask for hidden symmetry. Let us strip away that mask.

### The Modulus Trick

When you see the modulus squared of a complex number, your first instinct should always be the fundamental identity: $|z|^2 = z \cdot \bar{z}$. This is our most powerful tool. Since $a, b,$ and $c$ are integers, they are real.

When we take the conjugate of the entire expression, the bar only affects the complex parts, $\omega$ and $\omega^2$. Recall that for cube roots of unity, $\bar{\omega} = \omega^2$ and $\overline{\omega^2} = \omega$. This is a geometric necessity—they are reflections of each other across the real axis.

So, our expression becomes:

$$E = (a + b\omega + c\omega^2)(a + b\omega^2 + c\omega)$$

### The Algebraic Alchemy

Now, we expand. I know, expanding a trinomial multiplied by a trinomial sounds tedious, but stay with me. We are looking for patterns.

When you multiply these out, you get nine terms. You will see $a^2$, $b^2\omega^3$, $c^2\omega^3$, and various cross-terms like $ab\omega^2$, $ac\omega$, $ab\omega$, $bc\omega^2$, $ac\omega^2$, and $bc\omega$.

Here is where the magic happens. We know that $\omega^3 = 1$. This is the defining characteristic of our cube roots. Suddenly, those $b^2\omega^3$ and $c^2\omega^3$ terms simplify beautifully to just $b^2$ and $c^2$.

Our expression now looks like this:

$$E = a^2 + b^2 + c^2 + ab(\omega + \omega^2) + bc(\omega + \omega^2) + ca(\omega + \omega^2)$$

### The Elegant Simplification

Look at the term $(\omega + \omega^2)$ appearing everywhere. We know that $1 + \omega + \omega^2 = 0$. Therefore, $\omega + \omega^2 = -1$.

This is the moment where the complex numbers vanish entirely, leaving us with a purely real, symmetric polynomial:

$$E = a^2 + b^2 + c^2 - ab - bc - ca$$

This is a classic form in algebra. To find its minimum, we need to see it as a sum of squares. We multiply and divide by $2$ to create the perfect squares:

$$E = \frac{1}{2} [2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ca]$$

By rearranging, we get:

$$E = \frac{1}{2} [(a-b)^2 + (b-c)^2 + (c-a)^2]$$

### The Trap of Constraints

We have arrived at the heart of the problem. We want to minimize $E$. If there were no constraints, we would set $a=b=c$ to get $E=0$.

But the problem explicitly states that $a, b,$ and $c$ are **distinct non-zero integers**. This constraint is a gatekeeper. Because they are distinct, the differences $(a-b)$, $(b-c)$, and $(c-a)$ cannot be zero.

To minimize the sum of their squares, we must make these differences as small as possible. The smallest absolute difference between two distinct integers is $1$.

Let us test the simplest case: consecutive integers. Let $a=1, b=2, c=3$.

- $(a-b) = 1-2 = -1$
- $(b-c) = 2-3 = -1$
- $(c-a) = 3-1 = 2$

Substituting these into our formula:

$$E = \frac{1}{2} [(-1)^2 + (-1)^2 + (2)^2] = \frac{1}{2} [1 + 1 + 4] = \frac{1}{2} [6] = 3$$

### Conclusion

We have navigated the complex plane, utilized the properties of unity, performed algebraic manipulation, and respected the constraints of the problem. The **minimum value is 3**.

This problem teaches us that even when a problem looks like it belongs in the realm of complex analysis, the path to the solution often lies in the elegance of real algebra. Keep practicing, keep visualizing, and most importantly, keep falling in love with the process.

Animated Solution for Mathematics - Complex Numbers: Let $ω \neq = 1$ be a cube root of unity. Then the minimum of the set ${∣ a + bω + c ω^{2} ∣^{2} : a, b, c distinct non-zero integers}$ equals

Enter Numerical Value:

Visualized Solution

The Given Expression

Modulus Squared Property

Conjugates of $ω$

Expanding the Expression

Applying $ω^{3} = 1$

Grouping Terms

Sum of Roots of Unity

Simplified Expression

Sum of Squares Form

The Perfect Squares

Applying Constraints

Minimizing Differences

Evaluating the Minimum

Final Calculation

The Sigma Insight: Cube Roots and nth Roots of Unity

Analyzing the Setup

The Modulus Trick

The Algebraic Alchemy

The Elegant Simplification

The Trap of Constraints

Conclusion

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