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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

Note by Michael Tong
3 years, 12 months ago

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3 votes

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Actually, sophie-germain identity isn't very obscure. Well.. not as well known as difference of squares of difference of cubes though... Taehyung Kim · 3 years, 12 months ago

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See this. Akshat Sharda · 1 year, 10 months ago

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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 · 3 years, 12 months ago

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@Garrett Higginbotham 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 · 3 years, 12 months ago

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Seems like once I got this problem in Brilliant too... Bhargav Das · 3 years, 12 months ago

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@Bhargav Das Yes me too Arbër Avdullahu · 3 years, 11 months ago

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