Difference between revisions of "1992 OIM Problems/Problem 4"
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== Solution == | == 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. | * 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. | ||
Revision as of 20:16, 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
- 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.