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

Line 14: Line 14:
  
 
== Solution ==
 
== 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 ==
 
== See also ==
 
https://www.oma.org.ar/enunciados/ibe7.htm
 
https://www.oma.org.ar/enunciados/ibe7.htm

Revision as of 20:10, 19 December 2023

Problem

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

Solution

  • 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.

See also

https://www.oma.org.ar/enunciados/ibe7.htm