Is there a better way to solve this problem than using the Sophie-Germain Identity?

http://www.artofproblemsolving.com/Wiki/index.php/1987AIMEProblems/Problem_14

Evaluate $ \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)} $.

I'd be surprised if it was because usually MAA doesn't require you to know relatively obscure theorems in their competitions until you get to at least USAMO. Is there a more intuitive way to solve this problem?

Cheers

#Advice #MathProblem #Math

Easy Math Editor

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Comments

Actually, sophie-germain identity isn't very obscure. Well.. not as well known as difference of squares of difference of cubes though...

Taehyung Kim - 7 years, 9 months ago

Seems like once I got this problem in Brilliant too...

Bhargav Das - 7 years, 9 months ago

Yes me too

Arbër Avdullahu - 7 years, 9 months ago

Notice that all of the fourth powers differ be $12$ . Try using a substitution that takes advantage of this and see if you can somehow simplify the result.

Garrett Higginbotham - 7 years, 9 months ago

What I meant was, the fourth powers in the numerator all differ by $12$ and all the fourth powers in the denominator differ by $12$ .

Garrett Higginbotham - 7 years, 9 months ago

See this.

Akshat Sharda - 5 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$