24-th APMO,2002

Discussion on Asian Pacific Mathematical Olympiad (APMO)
User avatar
Masum
Posts:592
Joined:Tue Dec 07, 2010 1:12 pm
Location:Dhaka,Bangladesh
24-th APMO,2002

Unread post by Masum » Thu Mar 24, 2011 4:20 pm

Find all positive integers $a$ and $b$ such that $\frac {a^2 + b} {b^2 − a}$ and $\frac {b^2 + a} {a^2 − b}$ are both integers.
One one thing is neutral in the universe, that is $0$.

sourav das
Posts:461
Joined:Wed Dec 15, 2010 10:05 am
Location:Dhaka
Contact:

Re: 24-th APMO,2002

Unread post by sourav das » Sat Dec 10, 2011 10:46 pm

WLOG, Let, $a\geq b$ So,
\[b^2+a\geq a^{2}-b\Rightarrow a+b\geq (a+b)(a-b)\Rightarrow b+1\geq a\]
Case 1: $a=b$
\[b-1|b+1\Rightarrow b-1|2\]
So, $a=b=2$ or $3$ in this case.
Case 2: $a=b+1$
\[b^{2}-b-1|(b+1)^{2}+b\Rightarrow b^2-b-1|4b+2\]
but $b^{2}-b-1\equiv 1$(mod $2$)
So, \[b^{2}-b-1|2b+1\]
But$b^{2}-b-1> 2b+1$ for $b\geq 4$
$b=3$ doesn't hold the divisibility. But true for b=1,2

So in this case (a,b)=(2,1) or (3,2)

A question: How to show "not divisible" sign in Latex
You spin my head right round right round,
When you go down, when you go down down......
(-$from$ "$THE$ $UGLY$ $TRUTH$" )

User avatar
*Mahi*
Posts:1175
Joined:Wed Dec 29, 2010 12:46 pm
Location:23.786228,90.354974
Contact:

Re: 24-th APMO,2002

Unread post by *Mahi* » Sun Dec 11, 2011 2:07 am

Use \not command.
Example $\not |$ $\not = \not \leq$
Please read Forum Guide and Rules before you post.

Use $L^AT_EX$, It makes our work a lot easier!

Nur Muhammad Shafiullah | Mahi

Post Reply