Difference between revisions of "1992 OIM Problems/Problem 4"
Line 24: | Line 24: | ||
<math>a_n^2+b_n^2=(n^2+An)^2+(Bn+8)^2</math> | <math>a_n^2+b_n^2=(n^2+An)^2+(Bn+8)^2</math> | ||
− | <math>a_n^2+b_n^2=( | + | <math>a_n^2+b_n^2=n^4+2An^3+(A^2+B^2)n^2+16Bn+8^2=S^2</math> |
− | <math> | + | Let <math>S^2=(n^2+Kn+8)^2</math> |
+ | <math>S^2=n^4+2Kn^3+(16+K^2)n^2+16Kn+8^2</math> | ||
Revision as of 20:19, 19 December 2023
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
- 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 1 on this one. I don't remember what I did.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.