Difference between revisions of "1987 AIME Problems/Problem 14"
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== Problem == | == Problem == | ||
Compute | Compute | ||
− | < | + | <cmath>\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}.</cmath> |
− | == Solution == | + | == Solution 1 (Sophie Germain Identity) == |
− | The [[Sophie Germain Identity]] states that <math>a^4 + 4b^4</math> can be | + | The [[Sophie Germain Identity]] states that <math>a^4 + 4b^4</math> can be factored as <math>\left(a^2 + 2b^2 - 2ab\right)\left(a^2 + 2b^2 + 2ab\right).</math> Each of the terms is in the form of <math>x^4 + 324.</math> Using Sophie Germain, we get that |
+ | <cmath>\begin{align*} | ||
+ | x^4 + 324 &= x^4 + 4\cdot 3^4 \\ | ||
+ | &= \left(x^2 + 2 \cdot 3^2 - 2\cdot 3\cdot x\right)\left(x^2 + 2 \cdot 3^2 + 2\cdot 3\cdot x\right) \\ | ||
+ | &= (x(x-6) + 18)(x(x+6)+18), | ||
+ | \end{align*}</cmath> | ||
+ | so the original expression becomes | ||
+ | <cmath>\frac{[(10(10-6)+18)(10(10+6)+18)][(22(22-6)+18)(22(22+6)+18)]\cdots[(58(58-6)+18)(58(58+6)+18)]}{[(4(4-6)+18)(4(4+6)+18)][(16(16-6)+18)(16(16+6)+18)]\cdots[(52(52-6)+18)(52(52+6)+18)]},</cmath> which simplifies to <cmath>\frac{(10(4)+18)(10(16)+18)(22(16)+18)(22(28)+18)\cdots(58(52)+18)(58(64)+18)}{(4(-2)+18)(4(10)+18)(16(10)+18)(16(22)+18)\cdots(52(46)+18)(52(58)+18)}.</cmath> | ||
+ | Almost all of the terms cancel out! We are left with <math>\frac{58(64)+18}{4(-2)+18} = \frac{3730}{10} = \boxed{373}.</math> | ||
− | + | ~Azjps (Solution) | |
− | + | ~MRENTHUSIASM (Minor Reformatting) | |
− | + | == Solution 2 (Completing the Square and Difference of Squares) == | |
+ | In both the numerator and the denominator, each factor is of the form <math>N^4+324=N^4+18^2</math> for some positive integer <math>N.</math> | ||
+ | |||
+ | We factor <math>N^4+18^2</math> by completing the square, then applying the difference of squares: | ||
+ | <cmath>\begin{align*} | ||
+ | N^4+18^2&=\left(N^4+36N^2+18^2\right)-36N^2 \\ | ||
+ | &=\left(N^2+18\right)^2-(6N)^2 \\ | ||
+ | &=\left(N^2-6N+18\right)\left(N^2+6N+18\right) \\ | ||
+ | &=\left((N-3)^2+9\right)\left((N+3)^2+9\right). | ||
+ | \end{align*}</cmath> | ||
+ | The original expression now becomes <cmath>\frac{\left[(7^2+9)(13^2+9)\right]\left[(19^2+9)(25^2+9)\right]\left[(31^2+9)(37^2+9)\right]\left[(43^2+9)(49^2+9)\right]\left[(55^2+9)(61^2+9)\right]}{\left[(1^2+9)(7^2+9)\right]\left[(13^2+9)(19^2+9)\right]\left[(25^2+9)(31^2+9)\right]\left[(37^2+9)(43^2+9)\right]\left[(49^2+9)(55^2+9)\right]}=\frac{61^2+9}{1^2+9}=\boxed{373}.</cmath> | ||
+ | ~MRENTHUSIASM | ||
+ | |||
+ | == Solution 3 (Complex Numbers) == | ||
+ | In both the numerator and the denominator, each factor is of the form <math>N^4+324=N^4+18^2</math> for some positive integer <math>N.</math> | ||
+ | |||
+ | We factor <math>N^4+18^2</math> by solving the equation <math>N^4+18^2=0,</math> or <math>N^4=-18^2.</math> | ||
+ | |||
+ | Two solutions follow from here: | ||
+ | |||
+ | === Solution 3.1 (Polar Form) === | ||
+ | We rewrite <math>N</math> to the polar form <cmath>N=r(\cos\theta+i\sin\theta)=r\operatorname{cis}\theta,</cmath> where <math>r</math> is the magnitude of <math>N</math> such that <math>r\geq0,</math> and <math>\theta</math> is the argument of <math>N</math> such that <math>0\leq\theta<2\pi.</math> | ||
+ | |||
+ | By <b>De Moivre's Theorem</b>, we have <cmath>N^4=r^4\operatorname{cis}(4\theta)=18^2(-1),</cmath> from which | ||
+ | <ol style="margin-left: 1.5em;"> | ||
+ | <li><math>r^4=18^2,</math> so <math>r=3\sqrt2.</math></li><p> | ||
+ | <li><math>\begin{cases} | ||
+ | \begin{aligned} | ||
+ | \cos(4\theta) &= -1 \\ | ||
+ | \sin(4\theta) &= 0 | ||
+ | \end{aligned}, | ||
+ | \end{cases}</math> so <math>\theta=\frac{\pi}{4},\frac{3\pi}{4},\frac{5\pi}{4},\frac{7\pi}{4}.</math></li><p> | ||
+ | </ol> | ||
+ | By the <b>Factor Theorem</b>, we get | ||
+ | <cmath>\begin{align*} | ||
+ | N^4+18^2&=\biggl(N-3\sqrt2\operatorname{cis}\frac{\pi}{4}\biggr)\biggl(N-3\sqrt2\operatorname{cis}\frac{3\pi}{4}\biggr)\biggl(N-3\sqrt2\operatorname{cis}\frac{5\pi}{4}\biggr)\biggl(N-3\sqrt2\operatorname{cis}\frac{7\pi}{4}\biggr) \\ | ||
+ | &=\biggl[\biggl(N-3\sqrt2\operatorname{cis}\frac{\pi}{4}\biggr)\biggl(N-3\sqrt2\operatorname{cis}\frac{7\pi}{4}\biggr)\biggr]\biggl[\biggl(N-3\sqrt2\operatorname{cis}\frac{3\pi}{4}\biggr)\biggl(N-3\sqrt2\operatorname{cis}\frac{5\pi}{4}\biggr)\biggr] \\ | ||
+ | &=\left[(N-(3+3i))(N-(3-3i))\right]\left[(N-(-3+3i))(N-(-3-3i))\right] \\ | ||
+ | &=\left[((N-3)-3i)((N-3)+3i)\right]\left[((N+3)-3i)((N+3)+3i)\right] \\ | ||
+ | &=\left[(N-3)^2+9\right]\left[(N+3)^2+9\right]. | ||
+ | \end{align*}</cmath> | ||
+ | We continue with the last paragraph of Solution 2 to get the answer <math>\boxed{373}.</math> | ||
+ | |||
+ | ~MRENTHUSIASM | ||
+ | |||
+ | === Solution 3.2 (Rectangular Form) === | ||
+ | We rewrite <math>N</math> to the rectangular form <cmath>N=a+bi</cmath> for some real numbers <math>a</math> and <math>b.</math> | ||
+ | |||
+ | Note that <math>N^2=\pm18i,</math> so there are two cases: | ||
+ | <ol style="margin-left: 1.5em;"> | ||
+ | <li><math>N^2=18i</math><p> | ||
+ | We have | ||
+ | <cmath>\begin{align*} | ||
+ | (a+bi)^2&=18i \\ | ||
+ | a^2-b^2+2abi&=18i. | ||
+ | \end{align*}</cmath> | ||
+ | We need <math>\begin{cases} | ||
+ | \begin{aligned} | ||
+ | a^2-b^2 &= 0 \\ | ||
+ | 2ab &= 18 | ||
+ | \end{aligned}, | ||
+ | \end{cases}</math> from which <math>(a,b)=(3,3),(-3,-3),</math> or <math>N=3+3i,-3-3i.</math></li><p> | ||
+ | <li><math>N^2=-18i</math><p> | ||
+ | We have | ||
+ | <cmath>\begin{align*} | ||
+ | (a+bi)^2&=-18i \\ | ||
+ | a^2-b^2+2abi&=-18i. | ||
+ | \end{align*}</cmath> | ||
+ | We need <math>\begin{cases} | ||
+ | \begin{aligned} | ||
+ | a^2-b^2 &= 0 \\ | ||
+ | 2ab &= -18 | ||
+ | \end{aligned}, | ||
+ | \end{cases}</math> from which <math>(a,b)=(3,-3),(-3,3),</math> or <math>N=3-3i,-3+3i.</math></li><p> | ||
+ | </ol> | ||
+ | By the <b>Factor Theorem</b>, we get | ||
+ | <cmath>\begin{align*} | ||
+ | N^4+18^2&=(N-(3+3i))(N-(-3-3i))(N-(3-3i))(N-(-3+3i)) \\ | ||
+ | &=\left[(N-(3+3i))(N-(3-3i))\right]\left[(N-(-3+3i))(N-(-3-3i))\right] \\ | ||
+ | &=\left[((N-3)-3i)((N-3)+3i)\right]\left[((N+3)-3i)((N+3)+3i)\right] \\ | ||
+ | &=\left[(N-3)^2+9\right]\left[(N+3)^2+9\right]. | ||
+ | \end{align*}</cmath> | ||
+ | We continue with the last paragraph of Solution 2 to get the answer <math>\boxed{373}.</math> | ||
+ | |||
+ | ~MRENTHUSIASM | ||
+ | |||
+ | == Solution 4 == | ||
+ | We use Sophie Germain's Identity to rewrite the first couple of multiplicands in the numerator and denominator. By Sophie Germain's: <cmath>(10^4 + 324) = (10^4 + \cdot 4 \cdot 3^4) = (10^2 + 2 \cdot 10 \cdot 3 + 2 \cdot 3^2)(10^2 - 2 \cdot 10 \cdot 3 + 2 \cdot 3^2) = (178)(58)</cmath> <cmath>(22^4 + 324) = (22^2 + 2 \cdot 3 \cdot 22 + 2 \cdot 3^2)(22^2 - 2 \cdot 3 \cdot 22 + 2 \cdot 3^2) = (634)(370)</cmath> <cmath>(4^4 + 324) = (4^2 + 2 \cdot 4 \cdot 3 + 2 \cdot 3^2)(4^2 - 2 \cdot 4 \cdot 3 + 2 \cdot 3^2) = (58)(10)</cmath> <cmath>(16^4 + 324) = (16^2 + 2 \cdot 16 \cdot 3 + 2 \cdot 3^2)(16^2 - 2 \cdot 16 \cdot 3 + 2 \cdot 3^2) = (370)(178)</cmath> If we only had these terms, then the fraction would rewrite to <math>\frac{(58)(178)(370)(634)}{(10)(58)(178)(370)}</math>. However, we notice most of the terms cancel, leaving us only with the largest term in the numerator and the smallest term in the denominator (<math>\frac{634}{10}</math>). We hypothesize that this will happen with the fraction as a whole. Then we will only be left with the largest term in the numerator, which is <math>(58^2 + 2 \cdot 58 \cdot 3 + 2 \cdot 3^2) = 3730</math>. TThe fraction simplifies to <math>\frac{3730}{10} = \boxed{373}</math>. | ||
+ | |||
+ | ~ cxsmi | ||
+ | |||
+ | == Video Solution by OmegaLearn == | ||
+ | https://youtu.be/ZWqHxc0i7ro?t=1023 | ||
+ | |||
+ | ~ pi_is_3.14 | ||
+ | |||
+ | == Video Solution== | ||
+ | https://www.youtube.com/watch?v=yoOWcx2Otcw | ||
+ | |||
+ | ~Michael Penn | ||
== See also == | == See also == |
Latest revision as of 10:42, 12 April 2024
Contents
Problem
Compute
Solution 1 (Sophie Germain Identity)
The Sophie Germain Identity states that can be factored as
Each of the terms is in the form of
Using Sophie Germain, we get that
so the original expression becomes
which simplifies to
Almost all of the terms cancel out! We are left with
~Azjps (Solution)
~MRENTHUSIASM (Minor Reformatting)
Solution 2 (Completing the Square and Difference of Squares)
In both the numerator and the denominator, each factor is of the form for some positive integer
We factor by completing the square, then applying the difference of squares:
The original expression now becomes
~MRENTHUSIASM
Solution 3 (Complex Numbers)
In both the numerator and the denominator, each factor is of the form for some positive integer
We factor by solving the equation
or
Two solutions follow from here:
Solution 3.1 (Polar Form)
We rewrite to the polar form
where
is the magnitude of
such that
and
is the argument of
such that
By De Moivre's Theorem, we have from which
so
so
By the Factor Theorem, we get
We continue with the last paragraph of Solution 2 to get the answer
~MRENTHUSIASM
Solution 3.2 (Rectangular Form)
We rewrite to the rectangular form
for some real numbers
and
Note that so there are two cases:
We have
We need
from which
or
We have
We need
from which
or
By the Factor Theorem, we get
We continue with the last paragraph of Solution 2 to get the answer
~MRENTHUSIASM
Solution 4
We use Sophie Germain's Identity to rewrite the first couple of multiplicands in the numerator and denominator. By Sophie Germain's:
If we only had these terms, then the fraction would rewrite to
. However, we notice most of the terms cancel, leaving us only with the largest term in the numerator and the smallest term in the denominator (
). We hypothesize that this will happen with the fraction as a whole. Then we will only be left with the largest term in the numerator, which is
. TThe fraction simplifies to
.
~ cxsmi
Video Solution by OmegaLearn
https://youtu.be/ZWqHxc0i7ro?t=1023
~ pi_is_3.14
Video Solution
https://www.youtube.com/watch?v=yoOWcx2Otcw
~Michael Penn
See also
1987 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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