Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

(Created page with "== Problem == Let <math>(a_n)</math> and <math>(b_n)</math> be two sequences of integers that verify the following conditions: i. <math>a_0 = 0</math>, <math>b_0 = 8</math>...")
 
(→‎See also)
 
(16 intermediate revisions by the same user not shown)
Line 4: Line 4:
 
i. <math>a_0 = 0</math>, <math>b_0 = 8</math>
 
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 pair <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
 
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
  
 
== Solution ==
 
== 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>
 +
 
 +
<math>a_n^2+b_n^2=n^4+2An^3+(A^2+B^2)n^2+16Bn+8^2=S^2</math>
 +
 
 +
Let <math>S^2=(n^2+Kn+8)^2</math>
 +
 
 +
<math>S^2=n^4+2Kn^3+(16+K^2)n^2+16Kn+8^2</math>
 +
 
 +
From the coefficient in front of <math>n^3</math> we find <math>2K=2A</math> thus <math>A=K</math>
 +
 
 +
From the coefficient in front of <math>n</math> we find <math>16K=16B</math> thus <math>B=K</math>, and <math>A=B=K</math>
 +
 
 +
From the coefficient in front of <math>n^2</math> we have:
 +
 
 +
<math>(16+K^2)=(A^2+B^2)=(K^2+K^2)</math> therefore <math>K^2=16</math>, thus <math>K= \pm 4</math>, and <math>A=B=\pm 4</math>
 +
 
 +
Substituting we have:
 +
 
 +
<math>a_n=n^2\pm 4n</math>, and <math>b_n=\pm 4n+8</math>
 +
 
 +
Thus
 +
 
 +
<math>(a_{1992},b_{1992})=(1992^2 + 4\cdot 1992,4\cdot 1992+8)</math>, or <math>(1992^2 - 4\cdot 1992,-4\cdot 1992+8)</math>
 +
 
 +
<math>(a_{1992},b_{1992})=(3976032,7976)</math>, or <math>(3960096,-7960)</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 10 on this one.  I don't remember what I did.
 +
 
 +
~Tomas Diaz. ~orders@tomasdiaz.com
 +
 
 +
 
 +
{{alternate solutions}}
  
 
== See also ==
 
== See also ==
 +
[[OIM Problems and Solutions]]
 +
 
https://www.oma.org.ar/enunciados/ibe7.htm
 
https://www.oma.org.ar/enunciados/ibe7.htm

Latest revision as of 09:42, 23 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

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 $b_n=Bn+8$

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

$a_n^2+b_n^2=n^4+2An^3+(A^2+B^2)n^2+16Bn+8^2=S^2$

Let $S^2=(n^2+Kn+8)^2$

$S^2=n^4+2Kn^3+(16+K^2)n^2+16Kn+8^2$

From the coefficient in front of $n^3$ we find $2K=2A$ thus $A=K$

From the coefficient in front of $n$ we find $16K=16B$ thus $B=K$, and $A=B=K$

From the coefficient in front of $n^2$ we have:

$(16+K^2)=(A^2+B^2)=(K^2+K^2)$ therefore $K^2=16$, thus $K= \pm 4$, and $A=B=\pm 4$

Substituting we have:

$a_n=n^2\pm 4n$, and $b_n=\pm 4n+8$

Thus

$(a_{1992},b_{1992})=(1992^2 + 4\cdot 1992,4\cdot 1992+8)$, or $(1992^2 - 4\cdot 1992,-4\cdot 1992+8)$

$(a_{1992},b_{1992})=(3976032,7976)$, or $(3960096,-7960)$

  • 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 10 on this one. I don't remember what I did.

~Tomas Diaz. ~orders@tomasdiaz.com


Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

See also

OIM Problems and Solutions

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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1992_OIM_Problems/Problem_4&oldid=208216"