Difference between revisions of "1992 OIM Problems/Problem 4"

Revision as of 21:10, 19 December 2023

Let $(a_n)$ and $(b_n)$ be two sequences of integers that verify the following conditions:

i. $a_0 = 0$ , $b_0 = 8$

ii. For all $n \geq 0$ , $a_{n+2}=2a_{n+1}-a_{n}+2$ , $b_{n+2}=2b_{n+1}-b_{n}$

iii. $a_{n}^{2}+b_{n}^{2}$ is a perfect square for all $n\ge 0$

Find at least two values of pair $(a_{1992},b_{1992})$ .

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

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.

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 == Solution ==
-* Note.  I actually competed at this event in Venezuela when I was in High School representing Puerto Rico.  I think I got partial points on this one.  I don't remember what I did.  I will try to solve it again later.
+* 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.
-{{solution}}
+{{alternate solutions}}
 == See also ==
 https://www.oma.org.ar/enunciados/ibe7.htm