Problem:
Compute
(44+324)(164+324)(284+324)(404+324)(524+324)(104+324)(224+324)(344+324)(464+324)(584+324)
Solution:
Since 324=182=4⋅34, we may use the factorization
x4+4y4=x4+4x2y2+4y4−4x2y2=(x2+2y2)2−(2xy)2=[(x2+2y2)−2xy][(x2+2y2)+2xy]=[(x2−2xy+y2)+y2][(x2+2xy+y2)+y2]=[(x−y)2+y2][(x+y)2+y2]
which yields in our case
n4+324=[(n−3)2+9][(n+3)2+9]
In view of this, the given fraction can be written as
(12+9)(72+9)(132+9)(192+9)⋯(492+9)(552+9)(72+9)(132+9)(192+9)(252+9)⋯(552+9)(612+9)
which simplifies to 12+9612+9=103730=373.
The problems on this page are the property of the MAA's American Mathematics Competitions