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Prompt: Let $a, b,$ and $c$ be complex numbers such that $a+b+c = 3$ and $(a+b+c)^3 = a^3 + b^3 + c^3 + 24$. Let $\zeta$ be a non-real cube root of unity and let $w = a + b\zeta + c\zeta^2$. Evaluate the value of $|w| \cdot \sqrt{3}$.

We are given that $a+b+c = 3$ and $(a+b+c)^3 = a^3 + b^3 + c^3 + 24$. We also know that $\zeta$ is a non-real cube root of unity, so $\zeta^3 = 1$ and $1 + \zeta + \zeta^2 = 0$.



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

We are asked to find the value of $|w| \cdot \sqrt{3}$, where $w = a + b\zeta + c\zeta^2$. To start, let's find the value of $w^3$ by cubing both sides of the equation: $$w^3 = (a + b\zeta + c\zeta^2)^3.$$



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

We can expand the right-hand side of the equation using the binomial theorem: $$w^3 = a^3 + b^3\zeta^3 + c^3\zeta^6 + 3a^2b\zeta + 3a^2c\zeta^2 + 3ab^2\zeta^2 + 3ac^2\zeta + 3b^2c\zeta + 3bc^2\zeta^2 + 6abc\zeta^3.$$



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Since $\zeta^3 = 1$, we can simplify the expression: $$w^3 = a^3 + b^3 + c^3 + 3a^2b\zeta + 3a^2c\zeta^2 + 3ab^2\zeta^2 + 3ac^2\zeta + 3b^2c\zeta + 3bc^2\zeta^2 + 6abc.$$



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Now, let's factor out $a^3 + b^3 + c^3$ from the equation $(a+b+c)^3 = a^3 + b^3 + c^3 + 24$: $$(a+b+c)^3 - (a^3 + b^3 + c^3) = 24.$$ Expanding the left-hand side gives $$3a^2b + 3a^2c + 3ab^2 + 3ac^2 + 3b^2c + 3bc^2 + 6abc = 24.$$



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Notice that this expression is very similar to the terms in the expression for $w^3$. In fact, if we substitute $\zeta = 1$ into the expression for $w^3$, we get $$a^3 + b^3 + c^3 + 3a^2b + 3a^2c + 3ab^2 + 3ac^2 + 3b^2c + 3bc^2 + 6abc.$$ So, we can rewrite the expression for $w^3$ as $$w^3 = (a^3 + b^3 + c^3) + (3a^2b + 3a^2c + 3ab^2 + 3ac^2 + 3b^2c + 3bc^2 + 6abc)\zeta + (3a^2b + 3a^2c + 3ab^2 + 3ac^2 + 3b^2c + 3bc^2 + 6abc)\zeta^2.$$



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Using the fact that $3a^2b + 3a^2c + 3ab^2 + 3ac^2 + 3b^2c + 3bc^2 + 6abc = 24$, we can simplify the expression for $w^3$: $$w^3 = (a^3 + b^3 + c^3) + 24\zeta + 24\zeta^2.$$



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Since $1 + \zeta + \zeta^2 = 0$, we can rewrite the expression for $w^3$ as $$w^3 = (a^3 + b^3 + c^3) - 24.$$



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

We are given that $(a+b+c)^3 = a^3 + b^3 + c^3 + 24$, so we can substitute this into the expression for $w^3$: $$w^3 = (a+b+c)^3 - 48.$$ Since $a+b+c = 3$, we have $$w^3 = 27 - 48 = -21.$$



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate

Finally, we can find the value of $|w|$ by taking the cube root of $|w^3|$: $$|w| = \sqrt[3]{|-21|} = \sqrt[3]{21}.$$ So, the value of $|w| \cdot \sqrt{3}$ is $$|w| \cdot \sqrt{3} = \sqrt[3]{21} \cdot \sqrt{3} = \sqrt[3]{21 \cdot 3 \sqrt{3}} = \sqrt[3]{63} = \boxed{\sqrt[3]{63}}.$$



Incorrect - LLM is appropriate


Correct - LLM is appropriate


Incorrect - Python is appropriate


Correct - Python is appropriate