1987 AIME Problem 14

Question

Compute

\frac{\begin{gathered}(10^4+324)(22^4+324)\\{}\cdot(34^4+324)(46^4+324)\\{}\cdot(58^4+324)\end{gathered}}{\begin{gathered}(4^4+324)(16^4+324)\\{}\cdot(28^4+324)(40^4+324)\\{}\cdot(52^4+324)\end{gathered}}.

Answer

Use Sophie Germain’s identity with

324=4\cdot3^4

If

g(x)=x^2+6x+18,

rewrite

x^4+324

as

g(x-6)g(x)

Let

g(x)=x^2+6x+18.

Sophie Germain’s identity gives

x^4+324=g(x-6)g(x).

The numerator therefore supplies

g(4),g(10),\ldots,g(58),

while the denominator supplies

g(-2),g(4),\ldots,g(52).

Everything cancels except

\begin{aligned}\frac{g(58)}{g(-2)}&=\frac{58^2+6(58)+18}{4-12+18}\\&=\frac{3730}{10}=373.\end{aligned}

Problem 14 in other years