×

# 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
4 years, 2 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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} $$\boxed{123}$$

Sort by:

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

- 4 years, 2 months ago

See this.

- 2 years, 1 month 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.

- 4 years, 2 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 $$12$$.

- 4 years, 2 months ago

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

- 4 years, 2 months ago

Yes me too

- 4 years, 2 months ago