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Difference between revisions of "2023 AIME II Problems/Problem 8"

(→‎Solution 2 (Moduli))
m (→‎Solution 1)
 
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 +
==Problem==
 +
 +
Let <math>\omega = \cos\frac{2\pi}{7} + i \cdot \sin\frac{2\pi}{7},</math> where <math>i = \sqrt{-1}.</math> Find the value of the product<cmath>\prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right).</cmath>
 +
 
==Solution 1==
 
==Solution 1==
  
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\left( \omega^{3\left( 7 - k \right)} + \omega^{\left( 7 - k \right)} + 1 \right) \\
 
\left( \omega^{3\left( 7 - k \right)} + \omega^{\left( 7 - k \right)} + 1 \right) \\
 
& = 3 \cdot 2^3 \\
 
& = 3 \cdot 2^3 \\
−
& = \boxed{\textbf{(024) }}.
+
& = \boxed{\textbf{024}}.
 
\end{align*}
 
\end{align*}
 
</cmath>
 
</cmath>
Line 39: Line 43:
 
Then, our product is equal to
 
Then, our product is equal to
  
−
<cmath>z_0z_1z_2z_3z_4z_5z_6.</cmath>
+
<cmath>|z_0||z_1||z_2||z_3||z_4||z_5||z_6|.</cmath>
  
 
<math>z_0 = 0</math>, and we may observe that <math>z_x</math> and <math>z_{7-x}</math> are conjugates for any <math>x</math>, meaning that their magnitudes are the same. Thus, our product is
 
<math>z_0 = 0</math>, and we may observe that <math>z_x</math> and <math>z_{7-x}</math> are conjugates for any <math>x</math>, meaning that their magnitudes are the same. Thus, our product is
  
−
<cmath>3z_1^2z_2^2z_3^2</cmath>
+
<cmath>3|z_1|^2|z_2|^2|z_3|^2</cmath>
−
<cmath> = 3((\cos \frac{6\pi}{7} + \cos \frac{2\pi}{7} + 1)^2 + (\sin \frac{6\pi}{7} + \sin \frac{2\pi}{7})^2)</cmath>
+
<cmath> = 3\left((\cos \frac{6\pi}{7} + \cos \frac{2\pi}{7} + 1)^2 + (\sin \frac{6\pi}{7} + \sin \frac{2\pi}{7})^2\right)
−
<cmath>((\cos \frac{12\pi}{7} + \cos \frac{4\pi}{7} + 1)^2 + (\sin \frac{12\pi}{7} + \sin \frac{4\pi}{7})^2)</cmath>
+
\left((\cos \frac{12\pi}{7} + \cos \frac{4\pi}{7} + 1)^2 + (\sin \frac{12\pi}{7} + \sin \frac{4\pi}{7})^2\right)
−
<cmath>((\cos \frac{4\pi}{7} + \cos \frac{6\pi}{7} + 1)^2 + (\sin \frac{4\pi}{7} + \sin \frac{6\pi}{7})^2)</cmath>
+
\left((\cos \frac{4\pi}{7} + \cos \frac{6\pi}{7} + 1)^2 + (\sin \frac{4\pi}{7} + \sin \frac{6\pi}{7})^2\right)</cmath>
  
 
Let us simplify the first term. Expanding, we obtain
 
Let us simplify the first term. Expanding, we obtain
Line 62: Line 66:
 
<cmath>3 + 2(\cos \frac{2\pi}{7} + \cos \frac{4\pi}{7} + \cos \frac{6\pi}{7}).</cmath>
 
<cmath>3 + 2(\cos \frac{2\pi}{7} + \cos \frac{4\pi}{7} + \cos \frac{6\pi}{7}).</cmath>
  
−
similarly, the second and third terms are, respectively,  
+
Similarly, the second and third terms are, respectively,  
  
 
<cmath>3 + 2(\cos \frac{4\pi}{7} + \cos \frac{8\pi}{7} + \cos \frac{12\pi}{7}),\textrm{ and}</cmath>
 
<cmath>3 + 2(\cos \frac{4\pi}{7} + \cos \frac{8\pi}{7} + \cos \frac{12\pi}{7}),\textrm{ and}</cmath>
 
<cmath>3 + 2(\cos \frac{6\pi}{7} + \cos \frac{12\pi}{7} + \cos \frac{4\pi}{7}).</cmath>
 
<cmath>3 + 2(\cos \frac{6\pi}{7} + \cos \frac{12\pi}{7} + \cos \frac{4\pi}{7}).</cmath>
 +
 +
Next, we have <math>\cos \frac{2\pi}{7} + \cos \frac{4\pi}{7} + \cos \frac{6\pi}{7} = -\frac{1}{2}</math>. This is because
 +
 +
<cmath>\cos \frac{2\pi}{7} + \cos \frac{4\pi}{7} + \cos \frac{6\pi}{7} = \frac{1}{2}(\textrm{cis }\frac{2\pi}{7} + \textrm{cis }\frac{4\pi}{7} + \textrm{cis }\frac{6\pi}{7} + \textrm{cis }\frac{8\pi}{7} + \textrm{cis }\frac{10\pi}{7} + \textrm{cis }\frac{12\pi}{7})</cmath>
 +
 +
<cmath> = \frac{1}{2}(-1)</cmath>
 +
<cmath> = -\frac{1}{2}.</cmath>
 +
 +
Therefore, the first term is simply <math>2</math>. We have <math>\cos x = \cos 2\pi - x</math>, so therefore the second and third terms can both also be simplified to <math>3 + 2(\cos \frac{2\pi}{7} + \cos \frac{4\pi}{7} + \cos \frac{6\pi}{7}) = 2</math>. Thus, our answer is simply
 +
 +
<cmath>3 \cdot 2 \cdot 2 \cdot 2</cmath>
 +
<cmath> = \boxed{\mathbf{024}}.</cmath>
 +
 +
~mathboy100
 +
 +
 +
==Solution 3 (Inspecting the exponents of powers of <math>\omega</math>)==
 +
We write out the product in terms of <math>\omega</math>:
 +
<cmath>\prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right)=3(\omega^3+\omega+1)(\omega^6+\omega^2+1)(\omega^9+\omega^3+1)(\omega^{12}+\omega^4+1)(\omega^{15}+\omega^5+1)(\omega^{18}+\omega^6+1).</cmath>
 +
 +
Grouping the terms in the following way exploits the fact that <math>\omega^{7k}=1</math> for an integer <math>k</math>, when multiplying out two adjacent products from left to right:
 +
 +
<cmath>\frac{1}{3} \prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right)=(\omega^3+\omega+1)(\omega^{18}+\omega^6+1)(\omega^6+\omega^2+1)(\omega^{15}+\omega^5+1)(\omega^9+\omega^3+1)(\omega^{12}+\omega^4+1).</cmath>
 +
 +
 +
When multiplying two numbers with like bases, we add the exponents. We can now rewrite the exponents of each product (two at a time, where <math>1</math> is treated as the identity) as a series of arrays:
 +
 +
<cmath> \textbf{(A)}\begin{bmatrix}
 +
3&1 &0 \\
 +
18&6&0\\
 +
\end{bmatrix}</cmath>
 +
 +
<cmath>\textbf{(B)}\begin{bmatrix}
 +
6&2 &0 \\
 +
15&5&0\\
 +
\end{bmatrix}</cmath>
 +
 +
<cmath>\textbf{(C)}\begin{bmatrix}
 +
9&3 &0 \\
 +
12&4&0\\
 +
\end{bmatrix}.</cmath>
 +
 +
 +
Note that <math>\omega=e^{\frac{2\pi i}{7}}</math>. When raising <math>\omega</math> to a power, the numerator of the fraction is <math>2</math> times whatever power <math>\omega</math> is raised to, multiplied by <math>\pi i</math>. Since the period of <math>\omega</math> is <math>2\pi,</math> we multiply each array by <math>2</math> then reduce each entry <math>\mod{14},</math> as each entry in an array represents an exponent which <math>\omega</math> is raised to.
 +
 +
 +
<cmath> \textbf{(A)}\begin{bmatrix}
 +
6&2 &0 \\
 +
8&12&0\\
 +
\end{bmatrix}</cmath>
 +
 +
<cmath>\textbf{(B)}\begin{bmatrix}
 +
12&4 &0 \\
 +
2&10&0\\
 +
\end{bmatrix}</cmath>
 +
 +
<cmath>\textbf{(C)}\begin{bmatrix}
 +
4&6 &0 \\
 +
10&8&0\\
 +
\end{bmatrix}.</cmath>
 +
 +
To obtain the correct exponents, we seperately add each element of the lower row to one element of the top row.
 +
 +
Therefore (after reducing <math>\mod 14</math> again), we get the following sets:
 +
 +
<cmath>\textbf{(A)}\ \{0, 4, 6, 10, 0, 2, 8, 12, 0\}</cmath>
 +
<cmath>\textbf{(B)}\ \{0, 8, 12, 6, 0, 4, 2, 10, 0\}</cmath>
 +
<cmath>\textbf{(C)}\ \{0, 12, 4, 2, 0, 8, 10, 8, 0\}.</cmath>
 +
 +
Raising <math>\omega</math> to the power of each element in every set then multiplying over <math>\textbf{(A)}, \textbf{(B)},</math> and <math>\textbf{(C)}</math> yields
 +
 +
<cmath>\frac{1}{3} \prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right)=\left(\sum_{a\in \textbf{(A)}} \omega^a\right)\left(\sum_{b\in \textbf{(B)}} \omega^b\right)\left(\sum_{c\in \textbf{(C)}} \omega^c\right)</cmath>
 +
 +
<cmath>=\left(\sum_{a\in \textbf{(A)}} \omega^a\right)^3</cmath>
 +
 +
<cmath>=\left(\omega^0+\omega^4+\omega^6+\omega^{10}+\omega^0+\omega^2+\omega^8+\omega^{12}+\omega^0\right)^3</cmath>
 +
 +
<cmath>=\left(3+\omega^2+\omega^4+\omega^6+\omega^8+\omega^{10}+\omega^{12}\right)^3,</cmath>
 +
as these sets are all identical.
 +
 +
Summing as a geometric series,
 +
 +
<cmath>\frac{1}{3} \prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right)=\left(3+\frac{\omega^2(\omega^{12}-1)}{\omega^2-1}\right)^3</cmath>
 +
 +
<cmath>=\left(3+\frac{\omega^{14}-\omega^2}{\omega^2-1}\right)^3</cmath>
 +
 +
<cmath>=\left(3+\frac{1-\omega^2}{\omega^2-1}\right)^3</cmath>
 +
 +
<cmath>=(3-1)^3=8.</cmath>
 +
 +
Therefore,
 +
 +
<cmath>\frac{1}{3} \prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right)=8,</cmath>
 +
and
 +
<cmath>\prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right)=3\cdot8=\boxed{\textbf{(024)}}.</cmath>
 +
 +
-Benedict T (countmath1)
 +
 +
 +
==Solution 4==
 +
 +
The product can be factored into <math>-(r-1)(s-1)(t-1)(r-w)(s-w)(t-w)(r-w^2)(s-w^2)(t-w^2)....(r-w^6)(s-w^6)(t-w^6)</math>,
 +
 +
 +
where <math>r,s,t</math> are the roots of the polynomial <math>x^3+x+1=0</math>.
 +
 +
 +
This is then <math>-(r^7-1)(s^7-1)(t^7-1)</math> because <math>(r^7-1)</math> and <math>(r-1)(r-w)(r-w^2)...(r-w^6)</math> share the same roots.
 +
 +
 +
To find <math>-(r^7-1)(s^7-1)(t^7-1)</math>,
 +
 +
 +
Notice that <math>(r^7-1)=(r-1)(r^6+r^5+r^4+r^3+r^2+r+1)</math>. Since r satisfies <math>x^3+x+1=0</math>, <math>r^6+r^4+r^3=0</math>
 +
 +
 +
Substituting, you are left with <math>r^5+r^2+r+1</math>. This is <math>r^2(r^3+1)+r+1</math>, and after repeatedly substituting <math>r^3+x+1=0</math> you are left with <math>-2r^3</math>.
 +
 +
 +
So now the problem is reduced to finding <math>-(r-1)(s-1)(t-1)(-2r^3)(-2s^3)(-2t^3)=8(rst)^3(r-1)(s-1)(t-1)</math>, and vietas gives you the result of <math>\boxed{24}</math> -resources
 +
 +
 +
==Video Solution by The Power of Logic==
 +
https://youtu.be/o6w9t43GpJs?si=aoe-uM3m5AIwpz_H
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2023|num-b=7|num-a=9|n=II}}
 
{{AIME box|year=2023|num-b=7|num-a=9|n=II}}
 +
 +
[[Category:Intermediate Algebra Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 23:37, 31 January 2024

Problem

Let $\omega = \cos\frac{2\pi}{7} + i \cdot \sin\frac{2\pi}{7},$ where $i = \sqrt{-1}.$ Find the value of the product\[\prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right).\]

Solution 1

For any $k\in Z$, we have, \begin{align*} & \left( \omega^{3k} + \omega^k + 1 \right) \left( \omega^{3\left( 7 - k \right)} + \omega^{\left( 7 - k \right)} + 1 \right) \\ & = \omega^{3 \cdot 7} + \omega^{2k + 7} + \omega^{3k} + \omega^{-2k + 3 \cdot 7} + \omega^7 + \omega^k + \omega^{3\left( 7 - k \right)} + \omega^{\left( 7 - k \right)} + 1 \\ & = 1 + \omega^{2k} + \omega^{3k} + \omega^{-2k} + 1 + \omega^k + \omega^{-3k} + \omega^{-k} + 1 \\ & = 2 + \omega^{-3k} \sum_{j=0}^6 \omega^{j k} \\ & = 2 + \omega^{-3k} \frac{1 - \omega^{7 k}}{1 - \omega^k} \\ & = 2 . \end{align*} The second and the fifth equalities follow from the property that $\omega^7 = 1$.

Therefore, \begin{align*} \Pi_{k=0}^6 \left( \omega^{3k} + \omega^k + 1 \right) & = \left( \omega^{3 \cdot 0} + \omega^0 + 1 \right) \Pi_{k=1}^3 \left( \omega^{3k} + \omega^k + 1 \right) \left( \omega^{3\left( 7 - k \right)} + \omega^{\left( 7 - k \right)} + 1 \right) \\ & = 3 \cdot 2^3 \\ & = \boxed{\textbf{024}}. \end{align*}

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Solution 2 (Moduli)

Because the answer must be a positive integer, it is just equal to the modulus of the product. Define $z_n = \left(\textrm{cis }\frac{2n\pi}{7}\right)^3 + \textrm{cis }\frac{2n\pi}{7} + 1$.

Then, our product is equal to

\[|z_0||z_1||z_2||z_3||z_4||z_5||z_6|.\]

$z_0 = 0$, and we may observe that $z_x$ and $z_{7-x}$ are conjugates for any $x$, meaning that their magnitudes are the same. Thus, our product is

\[3|z_1|^2|z_2|^2|z_3|^2\] \[= 3\left((\cos \frac{6\pi}{7} + \cos \frac{2\pi}{7} + 1)^2 + (\sin \frac{6\pi}{7} + \sin \frac{2\pi}{7})^2\right) \left((\cos \frac{12\pi}{7} + \cos \frac{4\pi}{7} + 1)^2 + (\sin \frac{12\pi}{7} + \sin \frac{4\pi}{7})^2\right) \left((\cos \frac{4\pi}{7} + \cos \frac{6\pi}{7} + 1)^2 + (\sin \frac{4\pi}{7} + \sin \frac{6\pi}{7})^2\right)\]

Let us simplify the first term. Expanding, we obtain

\[\cos^2 \frac{6\pi}{7} + \cos^2 \frac{2\pi}{7} + 1 + 2\cos \frac{6\pi}{7}\cos \frac{2\pi}{7} + 2\cos \frac{6\pi}{7} + 2\cos \frac{2\pi}{7} + \sin^2 \frac{6\pi}{7} + \sin^2 \frac{2\pi}{7} + 2\sin \frac{6\pi}{7}\sin \frac{2\pi}{7}.\]

Rearranging and cancelling, we obtain

\[3 + 2\cos \frac{6\pi}{7} + 2\cos \frac{2\pi}{7} + 2\cos \frac{6\pi}{7}\cos \frac{2\pi}{7} + 2\sin \frac{6\pi}{7}\sin \frac{2\pi}{7}.\]

By the cosine subtraction formula, we have $2\cos \frac{6\pi}{7}\cos \frac{2\pi}{7} + 2\sin \frac{6\pi}{7}\sin \frac{2\pi}{7} = \cos \frac{6\pi - 2\pi}{7} = \cos \frac{4\pi}{7}$.

Thus, the first term is equivalent to

\[3 + 2(\cos \frac{2\pi}{7} + \cos \frac{4\pi}{7} + \cos \frac{6\pi}{7}).\]

Similarly, the second and third terms are, respectively,

\[3 + 2(\cos \frac{4\pi}{7} + \cos \frac{8\pi}{7} + \cos \frac{12\pi}{7}),\textrm{ and}\] \[3 + 2(\cos \frac{6\pi}{7} + \cos \frac{12\pi}{7} + \cos \frac{4\pi}{7}).\]

Next, we have $\cos \frac{2\pi}{7} + \cos \frac{4\pi}{7} + \cos \frac{6\pi}{7} = -\frac{1}{2}$. This is because

\[\cos \frac{2\pi}{7} + \cos \frac{4\pi}{7} + \cos \frac{6\pi}{7} = \frac{1}{2}(\textrm{cis }\frac{2\pi}{7} + \textrm{cis }\frac{4\pi}{7} + \textrm{cis }\frac{6\pi}{7} + \textrm{cis }\frac{8\pi}{7} + \textrm{cis }\frac{10\pi}{7} + \textrm{cis }\frac{12\pi}{7})\]

\[= \frac{1}{2}(-1)\] \[= -\frac{1}{2}.\]

Therefore, the first term is simply $2$. We have $\cos x = \cos 2\pi - x$, so therefore the second and third terms can both also be simplified to $3 + 2(\cos \frac{2\pi}{7} + \cos \frac{4\pi}{7} + \cos \frac{6\pi}{7}) = 2$. Thus, our answer is simply

\[3 \cdot 2 \cdot 2 \cdot 2\] \[= \boxed{\mathbf{024}}.\]

~mathboy100


Solution 3 (Inspecting the exponents of powers of $\omega$)

We write out the product in terms of $\omega$: \[\prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right)=3(\omega^3+\omega+1)(\omega^6+\omega^2+1)(\omega^9+\omega^3+1)(\omega^{12}+\omega^4+1)(\omega^{15}+\omega^5+1)(\omega^{18}+\omega^6+1).\]

Grouping the terms in the following way exploits the fact that $\omega^{7k}=1$ for an integer $k$, when multiplying out two adjacent products from left to right:

\[\frac{1}{3} \prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right)=(\omega^3+\omega+1)(\omega^{18}+\omega^6+1)(\omega^6+\omega^2+1)(\omega^{15}+\omega^5+1)(\omega^9+\omega^3+1)(\omega^{12}+\omega^4+1).\]


When multiplying two numbers with like bases, we add the exponents. We can now rewrite the exponents of each product (two at a time, where $1$ is treated as the identity) as a series of arrays:

\[\textbf{(A)}\begin{bmatrix} 3&1 &0 \\ 18&6&0\\ \end{bmatrix}\]

\[\textbf{(B)}\begin{bmatrix} 6&2 &0 \\ 15&5&0\\ \end{bmatrix}\]

\[\textbf{(C)}\begin{bmatrix} 9&3 &0 \\ 12&4&0\\ \end{bmatrix}.\]


Note that $\omega=e^{\frac{2\pi i}{7}}$. When raising $\omega$ to a power, the numerator of the fraction is $2$ times whatever power $\omega$ is raised to, multiplied by $\pi i$. Since the period of $\omega$ is $2\pi,$ we multiply each array by $2$ then reduce each entry $\mod{14},$ as each entry in an array represents an exponent which $\omega$ is raised to.


\[\textbf{(A)}\begin{bmatrix} 6&2 &0 \\ 8&12&0\\ \end{bmatrix}\]

\[\textbf{(B)}\begin{bmatrix} 12&4 &0 \\ 2&10&0\\ \end{bmatrix}\]

\[\textbf{(C)}\begin{bmatrix} 4&6 &0 \\ 10&8&0\\ \end{bmatrix}.\]

To obtain the correct exponents, we seperately add each element of the lower row to one element of the top row.

Therefore (after reducing $\mod 14$ again), we get the following sets:

\[\textbf{(A)}\ \{0, 4, 6, 10, 0, 2, 8, 12, 0\}\] \[\textbf{(B)}\ \{0, 8, 12, 6, 0, 4, 2, 10, 0\}\] \[\textbf{(C)}\ \{0, 12, 4, 2, 0, 8, 10, 8, 0\}.\]

Raising $\omega$ to the power of each element in every set then multiplying over $\textbf{(A)}, \textbf{(B)},$ and $\textbf{(C)}$ yields

\[\frac{1}{3} \prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right)=\left(\sum_{a\in \textbf{(A)}} \omega^a\right)\left(\sum_{b\in \textbf{(B)}} \omega^b\right)\left(\sum_{c\in \textbf{(C)}} \omega^c\right)\]

\[=\left(\sum_{a\in \textbf{(A)}} \omega^a\right)^3\]

\[=\left(\omega^0+\omega^4+\omega^6+\omega^{10}+\omega^0+\omega^2+\omega^8+\omega^{12}+\omega^0\right)^3\]

\[=\left(3+\omega^2+\omega^4+\omega^6+\omega^8+\omega^{10}+\omega^{12}\right)^3,\] as these sets are all identical.

Summing as a geometric series,

\[\frac{1}{3} \prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right)=\left(3+\frac{\omega^2(\omega^{12}-1)}{\omega^2-1}\right)^3\]

\[=\left(3+\frac{\omega^{14}-\omega^2}{\omega^2-1}\right)^3\]

\[=\left(3+\frac{1-\omega^2}{\omega^2-1}\right)^3\]

\[=(3-1)^3=8.\]

Therefore,

\[\frac{1}{3} \prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right)=8,\] and \[\prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right)=3\cdot8=\boxed{\textbf{(024)}}.\]

-Benedict T (countmath1)


Solution 4

The product can be factored into $-(r-1)(s-1)(t-1)(r-w)(s-w)(t-w)(r-w^2)(s-w^2)(t-w^2)....(r-w^6)(s-w^6)(t-w^6)$,


where $r,s,t$ are the roots of the polynomial $x^3+x+1=0$.


This is then $-(r^7-1)(s^7-1)(t^7-1)$ because $(r^7-1)$ and $(r-1)(r-w)(r-w^2)...(r-w^6)$ share the same roots.


To find $-(r^7-1)(s^7-1)(t^7-1)$,


Notice that $(r^7-1)=(r-1)(r^6+r^5+r^4+r^3+r^2+r+1)$. Since r satisfies $x^3+x+1=0$, $r^6+r^4+r^3=0$


Substituting, you are left with $r^5+r^2+r+1$. This is $r^2(r^3+1)+r+1$, and after repeatedly substituting $r^3+x+1=0$ you are left with $-2r^3$.


So now the problem is reduced to finding $-(r-1)(s-1)(t-1)(-2r^3)(-2s^3)(-2t^3)=8(rst)^3(r-1)(s-1)(t-1)$, and vietas gives you the result of $\boxed{24}$ -resources


Video Solution by The Power of Logic

https://youtu.be/o6w9t43GpJs?si=aoe-uM3m5AIwpz_H

See also

2023 AIME II (Problems • Answer Key • Resources)
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