1992 OIM Problems/Problem 4: Difference between revisions

Revision as of 20:16, 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

First we find the non-recursive form of this with unknown $a_1$ and $b_1$ :

$a_n=na_1+n(n+1))$ , and $b_n=nb_1-8(n-1)$

Let $A=a_1-1$ , and $B=b_1-8$

$a_n=n^2+An$ , and $b_n=Bn+8$

$a_n^2+b_n^2=(n^2+An)^2+(Bn+8)^2$

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 ==
+First we find the non-recursive form of this with unknown <math>a_1</math> and <math>b_1</math>:
+<math>a_n=na_1+n(n+1))</math>, and <math>b_n=nb_1-8(n-1)</math>
+Let <math>A=a_1-1</math>, and <math>B=b_1-8</math>
+<math>a_n=n^2+An</math>, and <math>b_n=Bn+8</math>
+<math>a_n^2+b_n^2=(n^2+An)^2+(Bn+8)^2</math>
 * 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.