1992 OIM Problems/Problem 4
Problem
Let and be two sequences of integers that verify the following conditions:
i. ,
ii. For all , ,
iii. is a perfect square for all
Find at least two values of pair .
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
First we find the non-recursive form of this with unknown and :
, and
Let , and
, and
Let
From the coefficient in front of we find thus
From the coefficient in front of we find thus , and
From the coefficient in front of we have:
therefore , thus , and
Substituting we have:
, and
Thus
, or
, or
- Note. I actually competed at this event in Venezuela when I was in High School representing Puerto Rico. I think I got like 2 or 3 points out of 10 on this one. I don't remember what I did.
~Tomas Diaz. ~orders@tomasdiaz.com
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.