1987 AIME Problems/Problem 14
Problem
Compute
![$\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}.$](http://latex.artofproblemsolving.com/d/f/5/df539142cf6bf608ee4e1681b24f4458ab51e4ca.png)
Solution
Sophie Germain, what a dank god. Sophie Germain, what a dank god. Sophie Germain, what a dank god. Sophie Germain, what a dank god. Sophie Germain, what a dank god. Sophie Germain, what a dank god. Sophie Germain, what a dank god. Sophie Germain, what a dank god. Sophie Germain, what a dank god. Sophie Germain, what a dank god. Sophie Germain, what a dank god. Sophie Germain, what a dank god.Sophie Germain, what a dank god. Sophie Germain, what a dank god. Sophie Germain, what a dank god. Sophie Germain, what a dank god.Sophie Germain, what a dank god. Sophie Germain, what a dank god. Sophie Germain, what a dank god. Sophie Germain, what a dank god.Sophie Germain, what a dank god. Sophie Germain, what a dank god. Sophie Germain, what a dank god. Sophie Germain, what a dank god.Sophie Germain, what a dank god. Sophie Germain, what a dank god. Sophie Germain, what a dank god. Sophie Germain, what a dank god.
The Sophie Germain Identity states that can be factorized as
. Each of the terms is in the form of
. Using Sophie-Germain, we get that
.
![$\frac{[(10(10-6)+18)(10(10+6)+18)][(22(22-6)+18)(22(22+6)+18)]\cdots[(58(58-6)+18)(58(58+6)+18)]}{[(4(4-6)+18)(4(4+6)+18)][(16(16-6)+18)(16(16+6)+18)]\cdots[(52(52-6)+18)(52(52+6)+18)]}$](http://latex.artofproblemsolving.com/4/d/f/4df1c85169d79080a1e4ad7cd3cb96f9c11ec0c7.png)
![$= \frac{(10(4)+18)(10(16)+18)(22(16)+18)(22(28)+18)\cdots(58(52)+18)(58(64)+18)}{(4(-2)+18)(4(10)+18)(16(10)+18)(16(22)+18)\cdots(52(46)+18)(52(58)+18)}$](http://latex.artofproblemsolving.com/4/b/d/4bdb1dff907fef36c98d06fdb044a05f81004d30.png)
Almost all of the terms cancel out! We are left with .
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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