Difference between revisions of "2024 AIME II Problems/Problem 13"
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+ | ==Solution 3== | ||
+ | |||
+ | Denote <math>r_j = e^{\frac{i 2 \pi j}{13}}</math> for <math>j \in \left\{ 0, 1, \cdots , 12 \right\}</math>. | ||
+ | |||
+ | Thus, for <math>\omega \neq 1</math>, <math>\left( \omega^0, \omega^1, \cdots, \omega^{12} \right)</math> is a permutation of <math>\left( r_0, r_1, \cdots, r_{12} \right)</math>. | ||
+ | |||
+ | We have | ||
+ | \begin{align*}\ | ||
+ | \Pi_{k = 0}^{12} \left( 2 - 2 \omega^k + \omega^{2k} \right) | ||
+ | & = \Pi_{k=0}^{12} \left( 1 + i - \omega^k \right) | ||
+ | \left( 1 - i - \omega^k \right) \\ | ||
+ | & = \Pi_{k=0}^{12} \left( \sqrt{2} e^{i \frac{\pi}{4}} - \omega^k \right) | ||
+ | \left( \sqrt{2} e^{-i \frac{\pi}{4}} - \omega^k \right) \\ | ||
+ | & = \Pi_{k=0}^{12} \left( \sqrt{2} e^{i \frac{\pi}{4}} - r_k \right) | ||
+ | \left( \sqrt{2} e^{-i \frac{\pi}{4}} - r_k \right) \\ | ||
+ | & = \left( | ||
+ | \Pi_{k=0}^{12} \left( \sqrt{2} e^{i \frac{\pi}{4}} - r_k \right) | ||
+ | \right) | ||
+ | \left( | ||
+ | \Pi_{k=0}^{12} \left( \sqrt{2} e^{-i \frac{\pi}{4}} - r_k \right) | ||
+ | \right) . \hspace{1cm} (1) | ||
+ | \end{align*} | ||
+ | The third equality follows from the above permutation property. | ||
+ | |||
+ | Note that <math>r_0, r_1, \cdots , r_{12}</math> are all zeros of the polynomial <math>z^{13} - 1</math>. | ||
+ | Thus, | ||
+ | <cmath> | ||
+ | \[ | ||
+ | z^{13} - 1 | ||
+ | = \Pi_{k=0}^{12} \left( z - r_k \right) . | ||
+ | \] | ||
+ | </cmath> | ||
+ | |||
+ | Plugging this into Equation (1), we get | ||
+ | \begin{align*} | ||
+ | (1) | ||
+ | & = \left( \left( \sqrt{2} e^{i \frac{\pi}{4}} \right)^{13} - 1 \right) | ||
+ | \left( \left( \sqrt{2} e^{-i \frac{\pi}{4}} \right)^{13} - 1 \right) \\ | ||
+ | & = \left( - 2^{13/2} e^{i \frac{\pi}{4}} - 1 \right) | ||
+ | \left( - 2^{13/2} e^{-i \frac{\pi}{4}} - 1 \right) \\ | ||
+ | & = 2^{13} + 1 + 2^{13/2} \cdot 2 \cos \frac{\pi}{4} \\ | ||
+ | & = 2^{13} + 1 + 2^7 \\ | ||
+ | & = 8321 . | ||
+ | \end{align*} | ||
+ | |||
+ | Therefore, the answer is <math>\boxed{\textbf{(321) }}</math>. | ||
+ | |||
+ | ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ||
+ | |||
+ | |||
+ | |||
+ | |||
+ | ==Solution 4== | ||
+ | |||
+ | |||
+ | Since <math>\omega \ne 1</math> is a <math>13^\text{th}</math> root of unity, and <math>13</math> is a prime, we have | ||
+ | <cmath> | ||
+ | f(z) \coloneqq z^{13} - 1 = \prod_{k = 0}^{12}(z - \omega^k) | ||
+ | </cmath> | ||
+ | by the Fundamental Theorem of Algebra. Next, observe that the quadratic <math>2 - 2z + z^2</math> factors as | ||
+ | <cmath> | ||
+ | 2 - 2z + z^2 = (1 - i - z)(1 + i - z). | ||
+ | </cmath> | ||
+ | Take the product of the above identity over <math>z \in \{1, \omega, \omega^2, \dots, \omega^{12} \}</math> to get the product of interest \begin{align*} | ||
+ | P &:= \prod_{k = 0}^{12}(2 - 2\omega^k + \omega^{2k}) \\ | ||
+ | &= \prod_{k = 0}^{12}(1 - i - \omega^k) \cdot \prod_{k = 0}^{12}(1 + i - \omega^k) \\ | ||
+ | &= f(1-i) \cdot f(1+i) \\ | ||
+ | &= \overline{f(1+i)} \cdot f(1+i) \\ | ||
+ | P &= \big| f(1+i) \big|^2. | ||
+ | \end{align*} | ||
+ | (Here, we use the fact that <math>f(\overline{z}) = \overline{f(z)}</math> whenever <math>f(z)</math> is a polynomial of real coefficients.) Next, notice that | ||
+ | <cmath> | ||
+ | (1+i)^{13} = (1+i)(1+i)^{12} = (1+i)\big( (1+i)^2 \big)^6 = (1+i)(2i)^6 = -64 - 64i | ||
+ | </cmath> | ||
+ | which means <math>f(1+i) = (1+i)^{13} - 1 = -65 - 64i</math>. So | ||
+ | <cmath> | ||
+ | P = \big| f(1+i) \big|^2 = \big| -65 - 64i \big|^2 = 65^2 + 64^2 = 8321 \equiv \boxed{321} \pmod{1000}. | ||
+ | </cmath> | ||
+ | And we are done. Alternatively, to add some geometric flavor, we can also compute <math>\big| f(1+i) \big|^2 = \big| (1+i)^{13} - 1 \big|^2</math> by law of cosines. | ||
+ | |||
+ | -- VensL. | ||
+ | |||
+ | |||
+ | |||
+ | ==Video Solution== | ||
+ | |||
+ | https://youtu.be/aSD8Xz0dAI8?si=PUDeOrRg-0bVXNpp | ||
+ | |||
+ | ~MathProblemSolvingSkills.com | ||
+ | |||
+ | ==Video Solution== | ||
+ | |||
+ | https://youtu.be/CtIdbP4F28Q | ||
+ | |||
+ | ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ||
==See also== | ==See also== |
Latest revision as of 10:36, 19 April 2024
Contents
Problem
Let be a 13th root of unity. Find the remainder when
is divided by 1000.
Solution 1
Now, we consider the polynomial whose roots are the 13th roots of unity. Taking our rewritten product from
to
, we see that both instances of
cycle through each of the 13th roots. Then, our answer is:
~Mqnic_
Solution 2
To find , where
and
, rewrite this is as
where
and
are the roots of the quadratic
.
Grouping the 's and
's results in
the denomiator by vietas.
the numerator by newtons sums
so the answer is
-resources
Solution 3
Denote for
.
Thus, for ,
is a permutation of
.
We have \begin{align*}\ \Pi_{k = 0}^{12} \left( 2 - 2 \omega^k + \omega^{2k} \right) & = \Pi_{k=0}^{12} \left( 1 + i - \omega^k \right) \left( 1 - i - \omega^k \right) \\ & = \Pi_{k=0}^{12} \left( \sqrt{2} e^{i \frac{\pi}{4}} - \omega^k \right) \left( \sqrt{2} e^{-i \frac{\pi}{4}} - \omega^k \right) \\ & = \Pi_{k=0}^{12} \left( \sqrt{2} e^{i \frac{\pi}{4}} - r_k \right) \left( \sqrt{2} e^{-i \frac{\pi}{4}} - r_k \right) \\ & = \left( \Pi_{k=0}^{12} \left( \sqrt{2} e^{i \frac{\pi}{4}} - r_k \right) \right) \left( \Pi_{k=0}^{12} \left( \sqrt{2} e^{-i \frac{\pi}{4}} - r_k \right) \right) . \hspace{1cm} (1) \end{align*} The third equality follows from the above permutation property.
Note that are all zeros of the polynomial
.
Thus,
Plugging this into Equation (1), we get \begin{align*} (1) & = \left( \left( \sqrt{2} e^{i \frac{\pi}{4}} \right)^{13} - 1 \right) \left( \left( \sqrt{2} e^{-i \frac{\pi}{4}} \right)^{13} - 1 \right) \\ & = \left( - 2^{13/2} e^{i \frac{\pi}{4}} - 1 \right) \left( - 2^{13/2} e^{-i \frac{\pi}{4}} - 1 \right) \\ & = 2^{13} + 1 + 2^{13/2} \cdot 2 \cos \frac{\pi}{4} \\ & = 2^{13} + 1 + 2^7 \\ & = 8321 . \end{align*}
Therefore, the answer is .
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Solution 4
Since is a
root of unity, and
is a prime, we have
by the Fundamental Theorem of Algebra. Next, observe that the quadratic
factors as
Take the product of the above identity over
to get the product of interest \begin{align*}
P &:= \prod_{k = 0}^{12}(2 - 2\omega^k + \omega^{2k}) \\
&= \prod_{k = 0}^{12}(1 - i - \omega^k) \cdot \prod_{k = 0}^{12}(1 + i - \omega^k) \\
&= f(1-i) \cdot f(1+i) \\
&= \overline{f(1+i)} \cdot f(1+i) \\
P &= \big| f(1+i) \big|^2.
\end{align*}
(Here, we use the fact that
whenever
is a polynomial of real coefficients.) Next, notice that
which means
. So
And we are done. Alternatively, to add some geometric flavor, we can also compute
by law of cosines.
-- VensL.
Video Solution
https://youtu.be/aSD8Xz0dAI8?si=PUDeOrRg-0bVXNpp
~MathProblemSolvingSkills.com
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See also
2024 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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