Difference between revisions of "1987 AIME Problems/Problem 14"

Revision as of 14:45, 2 July 2021

Problem

Compute $\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}.$

Solution 1 (Sophie Germain Identity)

The Sophie Germain Identity states that $a^4 + 4b^4$ can be factored as $(a^2 + 2b^2 - 2ab)(a^2 + 2b^2 + 2ab)$ . Each of the terms is in the form of $x^4 + 324$ . Using Sophie Germain, we get that $x^4 + 4\cdot 3^4 = (x^2 + 2 \cdot 3^2 - 2\cdot 3\cdot x)(x^2 + 2 \cdot 3^2 + 2\cdot 3\cdot x) = (x(x-6) + 18)(x(x+6)+18),$ so the original expression becomes

$\frac{[(10(10-6)+18)(10(10+6)+18)][(22(22-6)+18)(22(22+6)+18)]\cdots[(58(58-6)+18)(58(58+6)+18)]}{[(4(4-6)+18)(4(4+6)+18)][(16(16-6)+18)(16(16+6)+18)]\cdots[(52(52-6)+18)(52(52+6)+18)]}$

$= \frac{(10(4)+18)(10(16)+18)(22(16)+18)(22(28)+18)\cdots(58(52)+18)(58(64)+18)}{(4(-2)+18)(4(10)+18)(16(10)+18)(16(22)+18)\cdots(52(46)+18)(52(58)+18)}$

Almost all of the terms cancel out! We are left with $\frac{58(64)+18}{4(-2)+18} = \frac{3730}{10} = \boxed{373}$ .

Solution 2 (Completing the Square and Difference of Squares)

In both the numerator and the denominator, each factor is of the form $N^4+324=N^4+18^2$ for some positive integer $N.$

We factor $N^4+18^2$ by completing the square, then applying the difference of squares: $\begin{align*} N^4+18^2&=\left(N^4+36N^2+18^2\right)-36N^2 \\ &=\left(N^2+18\right)^2-(6N)^2 \\ &=\left(N^2-6N+18\right)\left(N^2+6N+18\right) \\ &=\left((N-3)^2+9\right)\left((N+3)^2+9\right). \end{align*}$ The original expression now becomes $\frac{\left[(7^2+9)(13^2+9)\right]\left[(19^2+9)(25^2+9)\right]\left[(31^2+9)(37^2+9)\right]\left[(43^2+9)(49^2+9)\right]\left[(55^2+9)(61^2+9)\right]}{\left[(1^2+9)(7^2+9)\right]\left[(13^2+9)(19^2+9)\right]\left[(25^2+9)(31^2+9)\right]\left[(37^2+9)(43^2+9)\right]\left[(49^2+9)(55^2+9)\right]}=\frac{61^2+9}{1^2+9}=\boxed{373}.$ ~MRENTHUSIASM

Solution 3 (Complex Numbers)

In both the numerator and the denominator, each factor is of the form $N^4+324=N^4+18^2$ for some positive integer $N.$

We factor $N^4+18^2$ by solving the equation $N^4+18^2=0,$ or $N^4=-18^2.$

Two solutions follow from here:

Solution 3.1 (Polar Form)

We rewrite $N$ to the polar form $N=r(\cos\theta+i\sin\theta)=r\mathrm{ \ cis \ }\theta,$ where $r$ is the magnitude of $N$ such that $r\geq0,$ and $\theta$ is the argument of $N$ such that $0\leq\theta<2\pi.$

By De Moivre's Theorem, we have $r^4\mathrm{cis}(4\theta)=18^2(-1),$ from which

$r^4=18^2,$ so $r=3\sqrt2.$

$\begin{cases} \begin{aligned} \cos(4\theta) &= -1 \\ \sin(4\theta) &= 0 \end{aligned}, \end{cases}$ so $\theta=\frac{\pi}{4},\frac{3\pi}{4},\frac{5\pi}{4},\frac{7\pi}{4}.$

By the Factor Theorem, we get $\begin{align*} N^4+18^2&=\biggl(N-3\sqrt2\mathrm{ \ cis \ }\frac{\pi}{4}\biggr)\biggl(N-3\sqrt2\mathrm{ \ cis \ }\frac{3\pi}{4}\biggr)\biggl(N-3\sqrt2\mathrm{ \ cis \ }\frac{5\pi}{4}\biggr)\biggl(N-3\sqrt2\mathrm{ \ cis \ }\frac{7\pi}{4}\biggr) \\ &=\biggl[\biggl(N-3\sqrt2\mathrm{ \ cis \ }\frac{\pi}{4}\biggr)\biggl(N-3\sqrt2\mathrm{ \ cis \ }\frac{7\pi}{4}\biggr)\biggr]\biggl[\biggl(N-3\sqrt2\mathrm{ \ cis \ }\frac{3\pi}{4}\biggr)\biggl(N-3\sqrt2\mathrm{ \ cis \ }\frac{5\pi}{4}\biggr)\biggr] \\ &=\left[(N-(3+3i))(N-(3-3i))\right]\left[(N-(-3+3i))(N-(-3-3i))\right] \\ &=\left[((N-3)-3i)((N-3)+3i)\right]\left[((N+3)-3i)((N+3)+3i)\right] \\ &=\left[(N-3)^2+9\right]\left[(N+3)^2+9\right]. \end{align*}$ We continue with the last paragraph of Solution 2 to get the answer $\boxed{337}.$

~MRENTHUSIASM

Solution 3.2 (Rectangular Form)

Video Solution

https://youtu.be/ZWqHxc0i7ro?t=1023

~ pi_is_3.14

@@ Line 31: / Line 31: @@
 Two solutions follow from here:
-=== Solution 3.1 (Exponential Form) ===
+=== Solution 3.1 (Polar Form) ===
-We rewrite <math>N</math> to the exponential form <cmath>N=re^{i\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>
+We rewrite <math>N</math> to the polar form <cmath>N=r(\cos\theta+i\sin\theta)=r\mathrm{ \ 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>r^4\mathrm{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\mathrm{ \ cis \ }\frac{\pi}{4}\biggr)\biggl(N-3\sqrt2\mathrm{ \ cis \ }\frac{3\pi}{4}\biggr)\biggl(N-3\sqrt2\mathrm{ \ cis \ }\frac{5\pi}{4}\biggr)\biggl(N-3\sqrt2\mathrm{ \ cis \ }\frac{7\pi}{4}\biggr) \\
+&=\biggl[\biggl(N-3\sqrt2\mathrm{ \ cis \ }\frac{\pi}{4}\biggr)\biggl(N-3\sqrt2\mathrm{ \ cis \ }\frac{7\pi}{4}\biggr)\biggr]\biggl[\biggl(N-3\sqrt2\mathrm{ \ cis \ }\frac{3\pi}{4}\biggr)\biggl(N-3\sqrt2\mathrm{ \ 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{337}.</math>
+~MRENTHUSIASM
 === Solution 3.2 (Rectangular Form) ===

1987 AIME (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
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