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Difference between revisions of "2003 AIME II Problems/Problem 15"

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== Problem ==
 
== Problem ==
−
Let
+
Let <cmath>P(x) = 24x^{24} + \sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).</cmath> Let <math>z_{1},z_{2},\ldots,z_{r}</math> be the distinct zeros of <math>P(x),</math> and let <math>z_{k}^{2} = a_{k} + b_{k}i</math> for <math>k = 1,2,\ldots,r,</math> where <math>a_{k}</math> and <math>b_{k}</math> are real numbers. Let
−
<center><math>P(x) = 24x^{24} + \sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).</math></center>
 
−
Let <math>z_{1},z_{2},\ldots,z_{r}</math> be the distinct zeros of <math>P(x),</math> and let <math>z_{K}^{2} = a_{k} + b_{k}i</math> for <math>k = 1,2,\ldots,r,</math> where <math>i = \sqrt { - 1},</math> and <math>a_{k}</math> and <math>b_{k}</math> are real numbers. Let
 
 
<center><math>\sum_{k = 1}^{r}|b_{k}| = m + n\sqrt {p},</math></center>
 
<center><math>\sum_{k = 1}^{r}|b_{k}| = m + n\sqrt {p},</math></center>
−
where <math>m,</math> <math>n,</math> and <math>p</math> are integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p.</math>
+
where <math>m, n,</math> and <math>p</math> are integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p.</math>
  
 
== Solution ==
 
== Solution ==
−
{{solution}}
+
This can be factored as:
 +
 
 +
<cmath> P(x) = x\left( x^{23} + x^{22} + \cdots + x^2 + x + 1 \right)^2 </cmath>
 +
 
 +
Note that <math> \left( x^{23} + x^{22} + \cdots + x^2 + x + 1 \right) \cdot (x-1) = x^{24} - 1 </math>.
 +
So the roots of <math>x^{23} + x^{22} + \cdots + x^2 + x + 1</math> are exactly all <math>24</math>-th complex roots of <math>1</math>, except for the root <math>x=1</math>.
 +
 
 +
Let <math>\omega=\cos \frac{360^\circ}{24} + i\sin \frac{360^\circ}{24}</math>. Then the distinct zeros of <math>P</math> are <math>0,\omega,\omega^2,\dots,\omega^{23}</math>.
 +
 
 +
We can clearly ignore the root <math>x=0</math> as it does not contribute to the value that we need to compute.
 +
 
 +
The squares of the other roots are <math>\omega^2,~\omega^4,~\dots,~\omega^{24}=1,~\omega^{26}=\omega^2,~\dots,~\omega^{46}=\omega^{22}</math>.
 +
 
 +
Hence we need to compute the following sum:
 +
 
 +
<cmath>R = \sum_{k = 1}^{23} \left|\, \sin \left( k\cdot \frac{360^\circ}{12} \right) \right|</cmath>
 +
 
 +
Using basic properties of the sine function, we can simplify this to
 +
 
 +
<cmath>R = 4 \cdot \sum_{k = 1}^{5} \sin \left( k\cdot \frac{360^\circ}{12} \right)</cmath>
 +
 
 +
The five-element sum is just <math>\sin 30^\circ + \sin 60^\circ + \sin 90^\circ + \sin 120^\circ + \sin 150^\circ</math>.
 +
We know that <math>\sin 30^\circ = \sin 150^\circ = \frac 12</math>, <math>\sin 60^\circ = \sin 120^\circ = \frac{\sqrt 3}2</math>, and <math>\sin 90^\circ = 1</math>.
 +
Hence our sum evaluates to:
 +
 
 +
<cmath>R = 4 \cdot \left( 2\cdot \frac 12 + 2\cdot \frac{\sqrt 3}2 + 1 \right) = 8 + 4\sqrt 3</cmath>
 +
 
 +
Therefore the answer is <math>8+4+3 = \boxed{015}</math>.
 +
 
 +
==Solution 2==
 +
 
 +
Note that <math>x^k + x^{k-1} + \dots + x + 1 = \frac{x^{k+1} - 1}{x - 1}</math>. Our sum can be reformed as <cmath>\frac{x(x^{47} - 1) + x^2(x^{45} - 1) + \dots + x^{24}(x - 1)}{x-1}</cmath>
 +
 
 +
So <cmath>\frac{x^{48} + x^{47} + x^{46} + \dots + x^{25} - x^{24} - x^{23} - \dots - x}{x-1} = 0</cmath>
 +
 
 +
<math>x(x^{47} + x^{46} + \dots - x - 1) = 0</math>
 +
 
 +
<math>x^{47} + x^{46} + \dots - x - 1 = 0</math>
 +
 
 +
<math>x^{47} + x^{46} + \dots + x + 1 = 2(x^{23} + x^{22} + \dots + x + 1)</math>
 +
 
 +
<math>\frac{x^{48} - 1}{x - 1} = 2\frac{x^{24} - 1}{x - 1}</math>
 +
 
 +
<math>x^{48} - 1 - 2x^{24} + 2 = 0</math>
 +
 
 +
<math>(x^{24} - 1)^2 = 0</math>
 +
 
 +
And we can proceed as above.
 +
 
 +
==Solution 3==
 +
 
 +
As in Solution 1, we find that the roots of <math>P(x)</math> we care about are the 24th roots of unity except <math>1</math>. Therefore, the squares of these roots are the 12th roots of unity. In particular, every 12th root of unity is counted twice, except for <math>1</math>, which is only counted once.
 +
 
 +
The possible imaginary parts of the 12th roots of unity are <math>0</math>, <math>\pm\frac{1}{2}</math>, <math>\pm\frac{\sqrt{3}}{2}</math>, and <math>\pm 1</math>. We can disregard <math>0</math> because it doesn't affect the sum.
 +
 
 +
<math>8</math> squares of roots have an imaginary part of <math>\pm\frac{1}{2}</math>, <math>8</math> squares of roots have an imaginary part of <math>\pm\frac{\sqrt{3}}{2}</math>, and <math>4</math> squares of roots have an imaginary part of <math>\pm 1</math>. Therefore, the sum equals <math>8\left(\frac{1}{2}\right) + 8\left(\frac{\sqrt{3}}{2}\right) + 4(1) = 8 + 4\sqrt{3}</math>.
 +
 
 +
The answer is <math>8+4+3=\boxed{015}</math>.
 +
 
 +
~rayfish
 +
 
 +
==Video Solution by Sal Khan==
 +
Part 1: https://www.youtube.com/watch?v=2eLAEMRrR7Q&list=PLSQl0a2vh4HCtW1EiNlfW_YoNAA38D0l4&index=3
 +
 
 +
Part 2: https://www.youtube.com/watch?v=TljVBB7gxbE
 +
 
 +
Part 3: https://www.youtube.com/watch?v=JTpXK2mENH4
 +
 
 +
- AMBRIGGS
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2003|n=II|num-b=14|after=Last Question}}
 
{{AIME box|year=2003|n=II|num-b=14|after=Last Question}}
 +
 +
[[Category: Intermediate Algebra Problems]]
 +
{{MAA Notice}}

Latest revision as of 16:23, 30 July 2022

Problem

Let \[P(x) = 24x^{24} + \sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).\] Let $z_{1},z_{2},\ldots,z_{r}$ be the distinct zeros of $P(x),$ and let $z_{k}^{2} = a_{k} + b_{k}i$ for $k = 1,2,\ldots,r,$ where $a_{k}$ and $b_{k}$ are real numbers. Let

$\sum_{k = 1}^{r}|b_{k}| = m + n\sqrt {p},$

where $m, n,$ and $p$ are integers and $p$ is not divisible by the square of any prime. Find $m + n + p.$

Solution

This can be factored as:

\[P(x) = x\left( x^{23} + x^{22} + \cdots + x^2 + x + 1 \right)^2\]

Note that $\left( x^{23} + x^{22} + \cdots + x^2 + x + 1 \right) \cdot (x-1) = x^{24} - 1$. So the roots of $x^{23} + x^{22} + \cdots + x^2 + x + 1$ are exactly all $24$-th complex roots of $1$, except for the root $x=1$.

Let $\omega=\cos \frac{360^\circ}{24} + i\sin \frac{360^\circ}{24}$. Then the distinct zeros of $P$ are $0,\omega,\omega^2,\dots,\omega^{23}$.

We can clearly ignore the root $x=0$ as it does not contribute to the value that we need to compute.

The squares of the other roots are $\omega^2,~\omega^4,~\dots,~\omega^{24}=1,~\omega^{26}=\omega^2,~\dots,~\omega^{46}=\omega^{22}$.

Hence we need to compute the following sum:

\[R = \sum_{k = 1}^{23} \left|\, \sin \left( k\cdot \frac{360^\circ}{12} \right) \right|\]

Using basic properties of the sine function, we can simplify this to

\[R = 4 \cdot \sum_{k = 1}^{5} \sin \left( k\cdot \frac{360^\circ}{12} \right)\]

The five-element sum is just $\sin 30^\circ + \sin 60^\circ + \sin 90^\circ + \sin 120^\circ + \sin 150^\circ$. We know that $\sin 30^\circ = \sin 150^\circ = \frac 12$, $\sin 60^\circ = \sin 120^\circ = \frac{\sqrt 3}2$, and $\sin 90^\circ = 1$. Hence our sum evaluates to:

\[R = 4 \cdot \left( 2\cdot \frac 12 + 2\cdot \frac{\sqrt 3}2 + 1 \right) = 8 + 4\sqrt 3\]

Therefore the answer is $8+4+3 = \boxed{015}$.

Solution 2

Note that $x^k + x^{k-1} + \dots + x + 1 = \frac{x^{k+1} - 1}{x - 1}$. Our sum can be reformed as \[\frac{x(x^{47} - 1) + x^2(x^{45} - 1) + \dots + x^{24}(x - 1)}{x-1}\]

So \[\frac{x^{48} + x^{47} + x^{46} + \dots + x^{25} - x^{24} - x^{23} - \dots - x}{x-1} = 0\]

$x(x^{47} + x^{46} + \dots - x - 1) = 0$

$x^{47} + x^{46} + \dots - x - 1 = 0$

$x^{47} + x^{46} + \dots + x + 1 = 2(x^{23} + x^{22} + \dots + x + 1)$

$\frac{x^{48} - 1}{x - 1} = 2\frac{x^{24} - 1}{x - 1}$

$x^{48} - 1 - 2x^{24} + 2 = 0$

$(x^{24} - 1)^2 = 0$

And we can proceed as above.

Solution 3

As in Solution 1, we find that the roots of $P(x)$ we care about are the 24th roots of unity except $1$. Therefore, the squares of these roots are the 12th roots of unity. In particular, every 12th root of unity is counted twice, except for $1$, which is only counted once.

The possible imaginary parts of the 12th roots of unity are $0$, $\pm\frac{1}{2}$, $\pm\frac{\sqrt{3}}{2}$, and $\pm 1$. We can disregard $0$ because it doesn't affect the sum.

$8$ squares of roots have an imaginary part of $\pm\frac{1}{2}$, $8$ squares of roots have an imaginary part of $\pm\frac{\sqrt{3}}{2}$, and $4$ squares of roots have an imaginary part of $\pm 1$. Therefore, the sum equals $8\left(\frac{1}{2}\right) + 8\left(\frac{\sqrt{3}}{2}\right) + 4(1) = 8 + 4\sqrt{3}$.

The answer is $8+4+3=\boxed{015}$.

~rayfish

Video Solution by Sal Khan

Part 1: https://www.youtube.com/watch?v=2eLAEMRrR7Q&list=PLSQl0a2vh4HCtW1EiNlfW_YoNAA38D0l4&index=3

Part 2: https://www.youtube.com/watch?v=TljVBB7gxbE

Part 3: https://www.youtube.com/watch?v=JTpXK2mENH4

- AMBRIGGS

See also

2003 AIME II (Problems • Answer Key • Resources)
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