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

Revision as of 23:47, 13 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 $(a_{1992},b_{1992})$ .

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

This problem needs a solution. If you have a solution for it, please help us out by adding it.

@@ Line 4: / Line 4: @@
 i. <math>a_0 = 0</math>, <math>b_0 = 8</math>
-ii. <math>a_1 = 2</math>
+ii. For all <math>n \geq 0</math>, <math>a_{n+2}=2a_{n+1}-a_{n}+2</math>, <math>b_{n+2}=2b_{n+1}-b_{n}</math>
-iii. <math>a_n</math> is a perfect square for all <math>n</math>
+iii. <math>a_{n}^{2}+b_{n}^{2}</math> is a perfect square for all <math>n\ge 0</math>
+Find at least two values of <math>(a_{1992},b_{1992})</math>.
-'''NOTE: my source made an error here and the text after the conditions is a problem from the previous year.  It's a mistake in my source below.  Therefore, this problem statement is incomplete.  If you can find the source for this problem please correct it and complete it'''
 ~translated into English by Tomas Diaz. ~orders@tomasdiaz.com